SUMS OF TWISTED GL(3) AUTOMORPHIC L-FUNCTIONS

by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein

Stanford University Boston College Brown University

This paper is dedicated to Professor Joseph Shalika

Abstract. Let F be a number field and π be an automorphic representation on GL_{r}(A_{F}). In this paper we consider weighted sums of quadratic twists of the Lfunction for π, ∑_{d} L(s, π, χ_{d}) a(s, π, d) Nd^{−w}, where χ_{d} is a quadratic character roughly attached to F (\sqrt{d})/F. We analyze the properties which the weights a(s, π, d) must satisfy if this is to satisfy a (certain, non-abelian) group of functional equations in (s, w), and show that there is a unique family of weight functions with this property for r ≤ 3. We describe these weights in detail when r = 3. As an application we give, for cuspidal π on GL_{3}(A_{ℚ}), a new proof of the holomorphicity of the symmetric square L-function of π. We also prove that if π′ is a cuspidal automorphic representation of GL(2) over ℚ then infinitely many quadratic twists of the adjoint square L-function of π′ are nonvanishing at the center of the critical strip.

0. Introduction. This paper concerns double Dirichlet series which may be expressed as weighted sums of quadratic twists of L-functions. In [BFH], a class of double Dirichlet series was proposed as follows. Let π be an automorphic representation of GL(r, A), where A is the adele ring of the global field F, let χ_{π} be the central character of π, and let ω be an idèle class character of A^{×}/F^{×}. Roughly

1991 Mathematics Subject Classification. Primary 11F66, Secondary 11F70, 11M41, 11N75. Key words and phrases. Automorphic representation, double Dirichlet series, quadratic twist,

twisted L-function, mean value.

Research supported in part by NSF grants DMS-9970841 (Bump), DMS-9970118 (Friedberg), and DMS-0088921 (Hoffstein).

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speaking, and restricting ourselves to the quadratic case, the Dirichlet series of interest is:

(0.1)  Z_{0}(s, w) = ∑_{d} L(s, π, χ_{d}) ω((d)) Nd^{−w}

where d runs through classes of F^{×} modulo squares, Nd is the absolute norm, χ_{d} is the quadratic character attached to F (\sqrt{d}), and L(s, π, χ_{d}) denotes the twisted L-function of π. As is explained in [BFH], the expectation is that this function of two variables will satisfy two functional equations generating a finite group if r ≤ 3 and an infinite group if r ≥ 4.

Unfortunately (0.1) is only an approximation to the actual Dirichlet series which we want. To explain why, let us sketch a method of studying such a series. First, each L-series being summed has a functional equation. Taking into account the power of the conductor of χ_{d} which occurs in the epsilon-factor for π ⊗ χ_{d}, one sees that there should be a functional equation for Z_{0}(s, w) under the transformation (s, w) → (1−s, w+r(s−1/2)). Second, writing L(s, π, χ_{d}) = ∑ c(m) χ_{d}(m) Nm^{−s}, where the sum is over a set of representatives for the nonzero integral ideals of F, the series Z_{0}(s, w) may be written

Z_{0}(s, w) = ∑_{d} ∑_{m} c(m) χ_{d}(m) ω((d)) Nm^{−s} Nd^{−w}.

Now if m is an integer, then quadratic reciprocity says roughly that χ_{d}(m) is equal to χ_{m}(d). Thus after an interchange of summation, the inner sum is roughly of the form ∑_{d} χ_{m}(d) ω((d)) Nd^{−w}, and this is roughly a GL_{1} L-function in w. This suggests that the series should have a second functional equation under the transformation (s, w) → (s + w − 1/2, 1 − w)! If this can be made rigorous, then for r ≤ 3 it is not hard to check that these two functional equations allow one to (meromorphically) continue Z_{0}(s, w) to a tube domain in ℂ^{2} whose convex hull is all of ℂ^{2}. Putting in a finite set of factors to cancel possible poles, one has a holomorphic function on this tube domain, and hence by the continuation theorem for functions of several complex variables which are holomorphic on a tube domain [Ho, Theorem 2.5.10], a function which continues to ℂ^{2}.

Now we can see the issues to be addressed. Indeed, for the interchange to yield a GL_{1} sum, one must sum over all d giving nonzero integral ideals of F, rather than over a set of representatives for F^{×}/(F^{×})^{2}. This is problematic for several reasons. First, it is natural to attach quadratic characters only to elements of F^{×}, not to ideals. In essence, issues of units and class number arise. In addition quadratic reciprocity must be formulated precisely. Third, if one is to sum over all d, this raises the question of how to get the desired functional equations. Even in the class-number-one case, the conductor of χ_{d} involves only the square-free-part of d. So if one is to sum over non-square-free d, one must include with the L-functions of (0.1) a second “correction factor” a(s, π, d) which also has a functional equation under s → 1 − s, this second functional equation giving the corresponding power of the square-part of d. In a similar way there must be such factors in w, after the interchange. Do such factors, behaving properly with respect to the functional equations and allowing the interchange of summation, exist? Are they unique?

The first two obstructions above, involving class number and quadratic reciprocity, have been resolved by Fisher and Friedberg [FF1, FF2]. We will describe their results briefly below. Then in this paper we will study these last questions by describing and analyzing the (essentially local) relations which must be satisfied by the correction factors for this sketch of proof to go through. Using this analysis, we will show that the last two questions may be answered in the affirmative for r ≤ 3—the correction factors do exist, and they are unique.

A different approach to the existence of the correction factors has recently been given by Fisher and Friedberg [FF1,FF2]; this approach does not consider the issue of their uniqueness. Correction factors also arise in the work of Siegel [Si] (r = 1) and Bump, Friedberg and Hoffstein [BFH] and Friedberg and Hoffstein [FH] (r = 2).

Let us now explain how to resolve the issues from algebraic number theory as in the work of Fisher and Friedberg. Let 𝔒 be the ring of integers of F, and let S be a finite set of places containing all even places and all archimedean places, and such that the ring of S-integers 𝔒_{S} has class number 1. (In the application to double Dirichlet series, we will suppose that π and ω are unramified outside S.) If T is any set of places, let I(T) denote the ideals of 𝔒 prime to T. Let Div(F) be the free abelian group generated by the places of F, and let C = ∑_{v∈S} n_{v}v ∈ Div(F), with n_{v} = ord_{v}(4) if v is even and n_{v} = 1 otherwise. Let H_{C} be the (narrow) ray class group modulo C, and let E_{0} ⊆ I(S) be a set of representatives for a basis for H_{C} ⊗ ℤ/2ℤ as a vector space over 𝔽_{2}. For each E_{0} ∈ E_{0} choose m_{E_{0}} ∈ F^{×} such that E_{0}𝔒_{S} = m_{E_{0}}𝔒_{S}. Let E ⊆ I(S) be the full set of representatives for H_{C} ⊗ ℤ/2ℤ obtained by taking all possible products of distinct elements of E_{0}. If E = ∏_{E_{0}∈E_{0}} E_{0}^{n_{E_{0}}} (each n_{E_{0}} = 0 or 1), then let m_{E} = ∏_{E_{0}∈E_{0}} m_{E_{0}}^{n_{E_{0}}} , so that E𝔒_{S} = m_{E}𝔒_{S}. Also, let (\frac{a}{∗}) be the power residue symbol (Jacobi symbol) attached to the extension F (\sqrt{a}) of F. Then one has

Proposition 0.1 (Fisher-Friedberg). Let I, I_{1} ∈ I(S) be coprime. Write I = (m)EG^{2} with E ∈ E, m ≡ 1 mod C and G ∈ I(S), (G, I_{1}) = 1. Then the quadratic power residue symbol (\frac{mm_{E}}{I_{1}} ) is defined. If I = (m′)E′G′^{2} is another such

decomposition, then E′ = E and (\frac{m′m_{E}}{I_{1}} ) = (\frac{mm_{E}}{I_{1}} ).

In view of this Proposition, let us define (following Fisher and Friedberg) the quadratic symbol (\frac{I}{I_{1}} ) by (\frac{I}{I_{1}} ) = (\frac{mm_{E}}{I_{1}} ) and the quadratic character χ_{I} by

χ_{I}(I_{1}) = (\frac{I}{I_{1}} ). This depends on the choices above, but we suppress this from the notation. Let S_{I} denote the support of the conductor of χ_{I}. One may check that if I = I′(I′′)^{2}, then χ_{I}(I_{1}) = χ_{I′}(I_{1}) whenever both are defined. This allows one to extend χ_{I} to a character of all ideals of I(S ∪ S_{I}).

Proposition 0.2 (Fisher-Friedberg). Reciprocity – Let I, I_{1} ∈ I(S) be disjoint, and α(I, I_{1}) = χ_{I}(I_{1})χ_{I_{1}}(I). Then α(I, I_{1}) depends only on the images of I and I_{1} in H_{C} ⊗ ℤ/2ℤ.

Using these results, let us reformulate our problem as follows. Let π be a cuspidal automorphic representation of GL_{r}(A) unramified outside S, and let ω be an idèle class character also unramified outside S, where S is as above. Regard ω as a character of I(S). For each d ∈ I(S), let L(s, π, χ_{d}) denote the partial L-function for π ⊗ χ_{d} with the places in S ∪ S_{d} removed. Then we will study the sum

(0.2)  Z(s, w; π, ω) = ∑_{d} L(s, π, χ_{d}) a(s, π, d) ω(d) Nd^{−w},

the sum over d ∈ I(S), where a(s, π, d) is a correction factor to be determined.

One may first ask how this object depends on the choices in the definition of the quadratic characters χ_{I} above, indeed whether or not this series is natural. As shown in Fisher-Friedberg, if one allows ω to vary over twists by characters of the finite group H_{C} ⊗ ℤ/2ℤ and similarly considers twists of π by such characters, then the resulting series span a finite dimensional vector space which is independent of all choices. Thus the problem is natural; moreover, one may establish the two functional equations for different bases of this vector space.

For x ∈ H_{C} ⊗ ℤ/2ℤ, let δ_{x} be the function of I(S) which projects to the characteristic function of x. For the first functional equation, a basis for the vector space above is given by

(0.3)  Z(s, w; π, δ_{x}ω) = ∑_{[d]∈x} L(s, π, χ_{d}) a(s, π, d) ω(d) Nd^{−w},

where the sum is over d ∈ I(S) projecting to x. It suffices to establish a functional equation for each such series. By the theory of GL_{r} L-functions, each L-function in (0.3) satisfies a functional equation of the form

(0.4)  L(s, π, χ_{d}) = ϵ(s, π ⊗ χ_{d}) L(1 − s, π̃, χ_{d}) ∏_{v∈S} \frac{L(1 − s, π̃_{v}, χ_{d,v})}{L(s, π_{v}, χ_{d,v})}.

Let E ∈ E represent x. Since d is in the class of x, we have d = (f)EG^{2} where f ∈ F^{×} satisfies f ≡ 1 mod C and where G ∈ I(S). Since f ≡ 1 mod C, f is a square in the local ring 𝔒_{v} for all v ∈ S. It follows that the local quadratic characters χ_{d,v} = χ_{fm_{E},v} = χ_{E,v} for all v ∈ S. Write d = d_{0}d_{1}^{2} with d_{0}, d_{1} ∈ I(S), d_{0} square-free. Recall that each epsilon-factor in (0.4) is of the form AB^{s−1/2} for some A, B. To obtain a functional equation for (0.3), we are concerned with the dependence of these quantities on d. It can be shown that the central value A for the twisted epsilon-factor in (0.4) is χ_{π}(d_{0}) times a quantity depending only on E. This may be established since outside S ∪ S_{d} the epsilon-factor is a product of GL_{1} epsilon-factors, and these may be computed by comparing Gauss sums. The remaining part of each epsilon factor, B^{s−1/2}, may also be studied by reducing to GL_{1}. One finds that the dependence of B on d is given by Nd_{0}^{−r}. Hence each L-function in (0.3) transforms under s → 1−s by χ_{π}(d_{0})Nd_{0}^{−r(s−1/2)} times a factor which may be pulled out of the sum. To achieve a functional equation for this sum, we must find a correction factor transforming by a suitable multiple of d_{1}. That is, we seek correction factors a satisfying the condition

(0.5)  a(s, π, d) = (Nd_{1})^{−2r(s−1/2)} χ_{π}(d_{1}^{2}) a(1 − s, π̃, d).

The correction factor is to be given by a finite Dirichlet polynomial, i.e. a polynomial in Np^{−s}, and the primes p must divide d_{1}. Equation (0.5) gives the first condition on these polynomials.

For the second functional equation, it is sufficient to study Z(s, w; πδ_{x}, ωδ_{y}) with x, y ∈ H_{C} ⊗ ℤ/2ℤ since (by the orthogonality relations) such series span the vector space of double Dirichlet series. Let L(s, π, χ_{d}) = ∑_{m} c(m) χ_{d}(m) Nm^{−s} where the sum is now over m ∈ I(S). Then after interchange of summation, one arrives at

(0.6) Z(s, w; πδ_{x}, ωδ_{y}) = ∑_{[m]∈x} ∑_{[d]∈y} c(m) χ_{d}(m) ω(d) a(s, π, d) Nm^{−s} Nd^{−w}.

Since [m] ∈ x, [d] ∈ y, we have χ_{d}(m) = α(y, x) χ_{m}(d) by Proposition 0.2. Thus the inner sum gives a sum of the GL_{1} L-functions L(w, ωρχ_{m}) (the characters ρ summing to give the characteristic function δ_{y}), provided the correction factors behave appropriately. More precisely, we seek to write (0.6) in the form

(0.7)  ∑ L(w, ωρχ_{m}) b(w, ωρ, π, m) Nm^{−s},

for certain (different) correction factors b(w, ωρ, π, m). (Here we have absorbed the coefficients c(m) into the correction factors b(w, ωρ, π, m).) Once again, since χ_{m} depends only on the square-free part of m, for this to be a sum of terms with a functional equation we must impose conditions on the b(w, ωρ, π, m). If m = m_{0}m_{1}^{2} with m_{0}, m_{1} ∈ I(S), m_{0} square-free, then we require

(0.8)  b(w, ωρ, π, m) = (Nm_{1})^{1−2w} ω(m_{1}^{2}) b(1 − w, ω^{−1}ρ, π, m).

(Note that ρ^{2} = 1.) The correction factor b(w, ωρ, π, m) is also to be given by a Dirichlet polynomial, this time in the primes dividing m_{1}. Equation (0.8) gives a condition on these polynomials.

In Section 1 we combine the two conditions (0.5), (0.8) and the relation among the a’s,b’s,π, and ω necessary for the interchange of summation to work, and analyze the resulting situation. We derive relations among the coefficients of the polynomials which give the correction factors, and show that these lead to a (very complicated) system of recursion relations upon the coefficients. This allows us to establish the uniqueness of the correction factor for r ≤ 3. Then in Section 2 we present a solution to the relations when r = 3.

In Section 3, we illustrate the utility of these double Dirichlet series by giving an application to the estimation of sums of twisted GL_{3} L-functions. In this section we restrict to F = ℚ and χ_{π}^{2} = 1 for convenience. We first give details of the continuation of Z(s, w; π, ω) and describe the precise pole. As (0.7) suggests, when ω^{2} = 1 there is a pole at w = 1 whose residue essentially gives the contribution from the terms where m is a square. This residue is (up to bad prime factors) precisely ζ(6s − 1) L(2s, π, ∨^{2}), where the last factor is the symmetric square L-function for π!

The analytic properties of the symmetric square L-function have already been studied by the Langlands-Shahidi method (Langlands [L] and Shahidi [Sh]) and by the Rankin-Selberg method (see Patterson and Piatetski-Shapiro [PP] and Bump and Ginzburg [BG]). The present method seems fundamentally different from either of these. We will prove:

Theorem 0.3. (i) Let f be an automorphic cuspidal representation of GL(3) over ℚ whose central character has square 1. Let M be the product of a finite set of primes including 2, ∞, and the primes dividing the level of f. Let L_{M}(s, f, ∨^{2}) denote the symmetric square L-series with the Euler factors corresponding to primes dividing M removed. Then ζ(3s − 1) L_{M}(s, f, ∨^{2}) is an analytic function of s for all s, with the possible exception of s = 1, where a pole of order 1 can occur, and at s = 2/3 where a pole comes from the zeta factor.

(ii) Restrict to the case when L_{M}(s, f, ∨^{2}) has a pole of order 1 at s = 1. Let d run through a sequence of discriminants such that d falls into a fixed quadratic residue class mod p for every prime dividing M (mod 8 if p = 2). Then there exist infinitely many d in this sequence such that L(1/2, f, χ_{d}) ≠ 0, where the L-series referred to is the twist of the L-series of f by the character χ_{d}.

From this we deduce:

Theorem 0.4. Let π′ be a cuspidal automorphic representation of GL(2) over ℚ. Then there exist infinitely many quadratic characters χ_{d} as in Theorem 0.3 (ii) such that L(1/2, π′, Ad^{2} ⊗ χ_{d}) ≠ 0.

Indeed, we may apply Theorem 0.3 (ii) to the adjoint (Gelbart-Jacquet [GJ]) lift π of π′ to GL(3). The symmetric square L-function of this is the Riemann zeta function times a twist of the symmetric fourth power L-function of π′. Now the (twisted) symmetric fourth power L-function is automorphic on GL(5) by Kim and Shahidi [KS], hence does not vanish at s = 1 by Jacquet and Shalika [JS1]. Therefore the symmetric square of π has a pole there and Theorem 0.3 is applicable.

Actually Theorem 0.3 (ii) is not more general than this special case. Indeed Ginzburg, Rallis and Soudry [GRS] have shown that if the symmetric or exterior square L-function of an automorphic form on GL(r) has a pole, then the automorphic form is a classical lift. When r = 3, this means that in order for the symmetric square L-function to have a pole, the automorphic cuspidal representation π of GL(3) must be a quadratic twist of the adjoint (Gelbart-Jacquet [GJ]) lift of an automorphic representation π′ of GL(2).

In Theorem 3.8 we establish a somewhat more refined mean value result from which Theorem 0.3 is derived. Another work which utilizes the results of this paper to give estimates for sums of L-functions is Diaconu, Goldfeld, and Hoffstein [DGH].

The authors wish to warmly thank Adrian Diaconu, Benji Fisher and Dorian Goldfeld for helpful conversations. We also wish to thank the Institute for Advanced Study and the Banker’s Trust Company Foundation for support. Finally, we are grateful to the referee for careful reading of the manuscript.

1. The interchange of summation. Let π be an automorphic representation of GL_{r}(A) and ω be an idèle class character, both unramified outside S. Recall that if d = d_{0}d_{1}^{2} with d_{0} square-free, then χ_{d} = χ_{d_{0}}. As above we write the partial L-function L(s, π, χ_{d}) = ∑_{m} c(m) χ_{d_{0}}(m) Nm^{−s}, where the sum is over m ∈ I(S) and we define χ_{d_{0}}(m) = 0 if d_{0} and m are not coprime. Suppose that the correction factor a(s, π, d) is given by a Dirichlet polynomial P_{d_{0},d_{1}}(s) in the primes dividing d_{1}. Then

(1.1) Z(s, w; π, ω) = ∑_{d=d_{0}d_{1}^{2} d_{0} square-free} ∑_{m} c(m) χ_{d_{0}}(m) Nm^{−s} P_{d_{0},d_{1}}(s) ω(d) Nd^{−w}.

As in Section 0, we seek polynomials P and Q, the latter giving the correction factors b(w, ωρ, π, m), such that

(1.2)

Z(s, w; π, ω) = ∑_{x,y} α(x, y) ∑_{m=m_{0}m_{1}^{2}∈[x] m_{0} square-free} ∑_{d∈[y]} ω(d) χ_{m_{0}}(d) Nd^{−w} Q_{m_{0},m_{1}}(w) Nm^{−s}.

Here the first sum is over x, y ∈ H_{C} ⊗ ℤ/2ℤ. We shall call the equality of (1.1) and (1.2) the basic identity. Denote the above quantity Z(s, w) for concision.

The degrees of the polynomials P, Q (if they exist) are determined by the relations (0.5), (0.8) respectively. We introduce the notation

P_{d_{0},d_{1}}(s) = ∏_{p^{α}||d_{1}}(1 + a_{d_{0},p}^{(α)} Np^{−s} + a_{d_{0},p^{2}}^{(α)} Np^{−2s} + … + a_{d_{0},p^{2rα}}^{(α)}Np^{−2rαs}),

Q_{m_{0},m_{1}}(w) =

∏_{p^{α}||m_{1}}(b_{m_{0},1}^{(α)} + ω(p) b_{m_{0},p}^{(α)} Np^{−w} + ω(p^{2}) b_{m_{0},p^{2}}^{(α)} Np^{−2w} + …

+ ω(p^{2α}) b_{m_{0},p^{2α}}^{(α)}Np^{−2αw}).

The coefficients of P here depend on π but are independent of ω by hypothesis, while those of Q depend on π, ω. (As we shall soon see, with this normalization of Q the coefficients b_{m_{0},p^{j}}^{(α)} turn out to be independent of ω.) We normalize the coefficients of P to begin with 1 as shown.

In this section we investigate the relations that the coefficients a_{d_{0},p^{j}}^{(α)} and b_{m_{0},p^{j}}^{(α)} must satisfy if they are to give the basic identity.

SUMS OF TWISTED GL(3) AUTOMORPHIC L-FUNCTIONS 9

Proposition 1.1. Let t_{0}, u_{0}, p ∈ I(S) be square-free with p prime and t_{0}, u_{0}, p pairwise coprime. Then the coefficients a_{d_{0},d_{1}}^{(α)} and b_{m_{0},m_{1}}^{(α)} must satisfy the following relations for all e, f ≥ 0:

(1.3)  c(t_{0}) ∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k+1} a_{u_{0},p^{2e+1−k}}^{(f)} = b_{pt_{0},p^{2f}}^{(e)} .
(1.4)  c(t_{0}) χ_{p}(t_{0}) a_{u_{0}p,p^{2e+1}}^{(f)} = χ_{u_{0}}(p) b_{t_{0}p,p^{2f+1}}^{(e)}.
(1.5)  c(t_{0}) [∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)} − a_{u_{0}p,p^{2e}}^{(f−1)}] = b_{t_{0},p^{2f}}^{(e)} .

(Here a_{u_{0}p,p^{2e}}^{(−1)} ≡ 0 by definition.)

(1.6)  c(t_{0}) χ_{p}(t_{0}) [a_{u_{0}p,p^{2e}}^{(f)} − ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}] = b_{t_{0},p^{2f+1}}^{(e)}.

In particular, for δ = 0, 1

b_{t_{0}p^{δ},p^{f}}^{(e)} = c(t_{0}) χ_{p}(t_{0})^{f} b_{p^{δ},p^{f}}^{(e)}
b_{t_{0}p^{δ},1}^{(e)} = c(t_{0}p^{δ+2e}),

and for all m_{0} ∈ I(S) the coefficients b_{m_{0},p^{f}}^{(e)} are independent of ω.

Note that the relations (1.3)–(1.6) are also consistent with the equation

a_{u_{0}p^{δ},p^{e}}^{(f)} = χ_{u_{0}}(p)^{e} a_{p^{δ},p^{e}}^{(f)} for δ = 0, 1.

Proof. We equate coefficients in the basic identity. Note that

(1.7)  α(ab, xy) = α(a, x)α(a, y)α(b, x)α(b, y).

Fix a prime p.

First, consider the coefficient in Z(s, w) of

Np^{−(2e+1)s−2fw} Nt_{0}^{−s} Nu_{0}^{−w} (e, f ≥ 0).

In (1.1), this arises from summands with (d_{0}, d_{1}) = (u_{0}, p^{f}). One must sum the contributions for m = t_{0}p^{k} with 0 ≤ k ≤ 2e + 1. In (1.2), this term arises when (m_{0}, m_{1}) = (pt_{0}, p^{e}). Thus in (1.2), we must have (d, p) = 1 (since otherwise χ_{pt_{0}}(d) = 0) so in fact d = u_{0}. Comparing terms we obtain

c(t_{0}) ω(u_{0}p^{2f}) χ_{u_{0}}(t_{0}) ∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(f)}

= α(pt_{0}, u_{0}) ω(u_{0}p^{2f}) χ_{pt_{0}}(u_{0}) b_{pt_{0},p^{2f}}^{(e)} .

Making use of Proposition 0.2, we have α(pt_{0}, u_{0}) χ_{pt_{0}}(u_{0}) = χ_{u_{0}}(pt_{0}). Canceling the factor ω(u_{0}p^{2f}) χ_{u_{0}}(t_{0}) from both sides, one obtains equation (1.3).

Second, consider the coefficient of

Np^{(2e+1)s−(2f+1)w} Nt_{0}^{−s} Nu_{0}^{−w} (e, f ≥ 0).

In (1.1), this arises from summands with (d_{0}, d_{1}) = (u_{0}p, p^{f}). In (1.2), it arises when (m_{0}, m_{1}) = (t_{0}p, p^{e}). This time, we must have (m, p) = 1 and hence m = t_{0} in (1.1), and (d, p) = 1 so d = u_{0} in (1.2). The factor χ_{u_{0}p}(t_{0}) appears in (1.1) and the factor α(t_{0}p, u_{0})χ_{t_{0}p}(u_{0}) = χ_{u_{0}}(t_{0}p) in (1.2). Canceling ω(u_{0}p^{2f+1}) χ_{u_{0}}(t_{0}), the relation obtained is simply (1.4). Later we will see that both sides of (1.4) are 0 for r ≤ 3.

Third, for the coefficients corresponding to an even power of Np^{−s}, it is best to multiply by a zeta factor in w first. Thus we consider the coefficient of

Np^{−2es−2fw} Nt_{0}^{−s} Nu_{0}^{−w} (e, f ≥ 0)

in

(1 − χ_{p}(t_{0}) ω(p) Np^{−w}) Z(s, w).

Substituting the expression for Z(s, w) from (1.1), we get contributions to this coefficient when (d_{0}, d_{1}) = (u_{0}, p^{f}) and when (d_{0}, d_{1}) = (u_{0}p, p^{f−1}). For this second term, one must have (m, p) = 1 so in that case m = t_{0}. On the other hand, substituting the expression from (1.2), one gets a contribution from (m_{0}, m_{1}) = (t_{0}, p^{e}), and then the sum telescopes! (Here one makes use of (1.7).) Canceling ω(u_{0}p^{2f}) χ_{u_{0}}(t_{0}), we arrive at the relation (1.5).

Fourth, consider the coefficient of

Np^{−2es−(2f+1)w} Nt_{0}^{−s} Nu_{0}^{−w} (e, f ≥ 0)

in (1 − χ_{p}(t_{0}) ω(p) Np^{−w})Z(s, w). Once again, substituting in the two sums for Z(s, w), the contributions in (1.1) arise from (d_{0}, d_{1}) = (u_{0}p, p^{f}) and (d_{0}, d_{1}) = (u_{0}, p^{f}). In the first of these one necessarily has m = t_{0}. The contributions in (1.2) arise from (m_{0}, m_{1}) = (t_{0}, p^{e}). Once again, this second sum telescopes. After canceling ω(u_{0}p^{2f+1}) χ_{u_{0}}(t_{0}), one obtains the relation (1.6).

The relations of Proposition 1.1 completely determine the b’s from the a’s. Surprisingly, they do more: they determine the a’s as well when r ≤ 3! Before explaining this, let us record the relations on the coefficients of P, Q implied by (0.5) and (0.8). These state that:

(1.8)  a_{d_{0},p^{i}}^{(α)} = χ_{π}(p)^{2α} Np^{i−rα}a_{d_{0},p^{2rα−i}}^{(α)},  0 ≤ i ≤ 2rα

(1.9)  b_{m_{0},p^{j}}^{(α)} = Np^{j−α} b_{m_{0},p^{2α−j}}^{(α)},  0 ≤ j ≤ 2α.

Here and below we let ã denote the family of coefficients associated with the contragredient representation π̃ of π. We also let c̃ be defined by L(s, π̃, χ_{d}) = ∑ c̃(m) χ_{d}(m) Nm^{−s}. (In the proof of (S2) below we assume that ã_{d_{0},p^{i}}^{(α)} = 0 im-

plies a_{d_{0},p^{i}}^{(α)} = 0. This is consistent with the formal equality of (1.1) and (1.2) for

all π which we are seeking.) Since ã_{d_{0},1}^{(α)} = 1 and b_{m_{0},1}^{(α)} = c(m_{0}p^{2α}), we deduce that

(1.10)  a_{d_{0},p^{2rα}}^{(α)} = χ_{π}(p)^{2α}Np^{rα},  b_{n_{0},p^{2α}}^{(α)} = Np^{α} c(m_{0}p^{2α}).

We turn now to the relations which follow from Proposition 1.1.

Theorem 1.2. Let u_{0}, p ∈ I(S) be square-free with p prime and (p, u_{0}) = 1. Then the coefficients a_{d_{0},p^{j}}^{(α)} must satisfy the following relations, which completely determine them if r ≤ 3.

First, the coefficients are stable in the sense that:

(S1) a_{u_{0},p^{e}}^{(f)} = a_{u_{0},p^{e}}^{(f′)} for f, f′ ≥ ⌊\frac{e+1}{2}⌋ .

(S2) a_{u_{0}p,p^{2e+1}}^{(f)} = 0 for f > 0 and e ≥ 0, if r ≤ 3.

(S3) a_{u_{0}p,p^{2e}}^{(f)} = a_{u_{0}p,p^{2e}}^{(f+1)} for f ≥ e.

Second, the coefficients satisfy the relations: (R1) a_{u_{0},p^{2e+1}}^{(f)} = − ∑_{k=1}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(f)} for f > e. (R2) a_{u_{0},p^{2e+1}}^{(e)} = χ_{u_{0}}(p) Np^{e} c(p^{2e+1}) − ∑_{k=1}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(e)} for e ≥ 1. (R3)

a_{u_{0},p^{2e+1}}^{(f)} = Np^{2f−e} ∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(e−f)} − ∑_{k=1}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(f)}

for f ≤ e < 2f. (R4) a_{u_{0},p^{2e}}^{(f)} = a_{u_{0}p,p^{2e}}^{(f−1)} − ∑_{k=1}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)} for f > e. (R5) a_{u_{0},p^{2e}}^{(e)} = Np^{e} c(p^{2e}) + a_{u_{0}p,p^{2e}}^{(e−1)} − ∑_{k=1}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(e)}. (R6)

a_{u_{0},p^{2e}}^{(f)} = a_{u_{0}p,p^{2e}}^{(f−1)} − ∑_{k=1}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}

+ Np^{2f−e} [ ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(e−f)} − a_{u_{0}p,p^{2e}}^{(e−f−1)}]

for f ≤ e < 2f. (R7) a_{u_{0}p,p^{2e}}^{(f)} = ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)} for f ≥ e. (R8)

a_{u_{0}p,p^{2e}}^{(f)} = ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}

+ Np^{2f+1−e}[a_{u_{0}p,p^{2e}}^{(e−f−1)} − ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(e−f−1)}]

for f < e < 2f + 1.

Proof. In this proof, we use the relations of Proposition 1.1 with t_{0} = 1 and the functional equation (1.9) extensively. The coefficients with even and odd indices will be treated separately, since they satisfy separate relations per Proposition 1.1.

First let us obtain stability and also some of the a’s. Observe that b_{m_{0},p^{2f}}^{(e)} = 0 for f > e (see equation (0.8) or (1.9)). Applying this to (1.5) we see that (since c(1) ≠ 0)

a_{u_{0}p,p^{2e}}^{(f−1)} = ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}

for f > e. Similarly, b_{1,p^{2f+1}}^{(e)} = 0 for f ≥ e, so (1.6) gives

a_{u_{0}p,p^{2e}}^{(f)} = ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}.

Thus we conclude that relation (R7) holds and also that stability (S3) holds. We also deduce that the sum

T (e, f) = ∑_{k=0}^{2e} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e−k}}^{(f)}

satisfies T (e, f) = T (e, f′) for f, f′ ≥ e.

Next we turn to relation (1.3). Once again the right side vanishes for f > e. For example, when e = 0 we obtain a_{u_{0},p}^{(f)} = −χ_{u_{0}}(p) c(p) for f > 0. We also deduce that

T_{o}(e, f) = ∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(f)}

satisfies T_{o}(e, f) = T_{o}(e, f′) for f, f′ > e.

We say that the coefficients a_{d_{0},p^{k}}^{(f)} are stable if they are independent of f for f sufficiently large (depending on k). To establish this, alternating between T_{o} and T, we obtain the stability of the a’s up to the top one in each sum. Since the entire expression is stable, the top term must be too. For example, since T (1, f) = T (1, f′) for f, f′ ≥ 1, we conclude that a_{u_{0},p^{2}}^{(f)} is stable for f ≥ 1. Then

since T_{o}(1, f) = T_{o}(1, f′) for f, f′ > 1 we conclude that a_{u_{0},p^{3}}^{(f)} is stable for f ≥ 2. Continuing in this way, we obtain stability relation (S1).

Next, relation (1.4) implies that

a_{u_{0}p,p^{2e+1}}^{(f)} = 0 f ≥ e,

b_{p,p^{2f+1}}^{(e)} = 0 e ≥ rf.

These quantities are then 0 for r ≤ 3. (This may be true in general but we do not check it here.) Indeed, if r = 1 this is immediate from the above, as b_{p,p^{2f+1}}^{(e)} = 0 for e ≥ f and for f ≥ e. For r ≥ 2, since b_{t_{0}p,p}^{(e)} = 0 for e ≥ 0, the functional equation (1.9) in the b’s implies that b_{t_{0}p,p^{2e−1}}^{(e)} = 0 for e ≥ 0. By relation (1.4), a_{u_{0}p,p^{2e+1}}^{(f)} = 0 for f = e − 1. Since a_{u_{0}p,p^{2e+1}}^{(f)} = 0 for e ≤ f + 1, the functional equation (1.8) in the a’s gives

a_{u_{0}p,p^{2e+1}}^{(f)} = 0 for \frac{2rf − (2e + 1) − 1}{2} ≤ f + 1.

This last condition may be rewritten as f(r − 1) ≤ e + 2. Thus by relation (1.4) once again,

b_{t_{0}p,p^{2f+1}}^{(e)} = 0 for 2f + 1 ≤ 2\frac{e + 2}{r − 1} + 1.

For r = 2, 3 this holds for 2f + 1 ≤ e. By the functional equation (1.9) it follows that all b_{t_{0}p,p^{2f+1}}^{(e)} = 0, hence by (1.4), all a_{u_{0}p,p^{2e+1}}^{(f)} = 0. This establishes stability condition (S2).

We now deduce the relations for the remaining relations (R) on the a’s.

First, relation (R1) is obtained from relation (1.3) since the right hand side is 0 when f > e. Similarly, relation (R2) is obtained from relation (1.3) with e = f; using b_{p,p^{2e}}^{(e)} = Np^{e} c(p^{2e+1}) (see (1.10)). As for (R3), we note that (1.9) gives

b_{p,p^{2f}}^{(e)} = Np^{2f−e} b_{p,p^{2e−2f}}^{(e)} .

Thus (1.3) gives

∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(f)} = Np^{2f−e} ∑_{k=0}^{2e+1} c(p^{k}) χ_{u_{0}}(p)^{k} a_{u_{0},p^{2e+1−k}}^{(e−f)}.

Moving the left hand summands with k ≥ 1 to the right hand side, relation (R3) follows. Note that the relation only determines a_{u_{0},p^{2e+1}}^{(e)} if 0 ≤ e − f < f or f ≤ e < 2f. (If e − f < 0 the right hand side vanishes and we get relation (R1).)

Similarly we obtain the relations for a_{u_{0},p^{2e}}^{(f)} from relation (1.5) above. If f > e the right side of (1.5) vanishes. Solving for the a_{u_{0},p^{2e}}^{(f)} gives precisely (R4). If e = f, the right side of (1.5) equals Np^{e} c(p^{2e}) and proceeding as above we get (R5). Finally to get (R6), we use the functional equation

b_{1,p^{2f}}^{(e)} = Np^{2f−e} b_{1,p^{2(e−f)}}^{(e)}.

Thus (1.5) gives precisely (R6) when one uses this, equating the expressions in the a’s corresponding to the above equation. Note that one gets information only when e − f < f and the indexing with e − f ≤ 0 is already captured in (R4), (R5) above.

Lastly, one determines the coefficients a_{u_{0}p,p^{2e}}^{(f)} from equation (1.6). If f ≥ e the right side of (1.6) equals zero, which immediately gives (R7), as already noted above. To get (R8), one may use the functional equation

b_{1,p^{2f+1}}^{(e)} = Np^{2f−e+1} b_{1,p^{2(e−f−1)+1}}^{(e)}.

Substituting the expressions which, by (1.6), correspond to each side of this equation and solving, one arrives at (R8). One needs 0 ≤ e−f −1 < f to extract information from this relation.

This concludes the proof of Theorem 1.2.

Note that none of these arguments above uses a specific value for the rank r, except for our treatment of the vanishing of a_{u_{0}p,p^{2e+1}}^{(f)} and b_{t_{0}p,p^{2f+1}}^{(e)}. However the relations given do not determine all other coefficients when r ≥ 4. In contrast, in the case r ≤ 3, the relations in Theorem 1.2 overdetermine the coefficients. It is not hard to check that if there is a set of solutions to the relations given in Proposition 1.1 and Theorem 1.2, then the basic identity does indeed hold.

2. The coefficients for r = 3.

Theorem 2.1. The combinatorial problem of Section 1 has a unique solution when r = 3.

Uniqueness has already been noted. To prove existence, we describe the coefficients explicitly. Another approach to the existence of the coefficients has recently been found by Fisher and Friedberg [FF2].

Fix a prime p and square-free u_{0} ∈ I(S) prime to p. Let α, β and γ be the Langlands parameters of π_{p}. Thus c(p^{k}) is the k-th complete symmetric polynomial in α, β and γ. More generally, let c(p^{k_{1}}, p^{k_{2}}) be the Schur polynomial

c(p^{k_{1}}, p^{k_{2}}) = \begin{matrix}| & \begin{matrix}α^{k_{1}+k_{2}+2} & β^{k_{1}+k_{2}+2} & γ^{k_{1}+k_{2}+2} \\ α^{k_{2}+1} & β^{k_{2}+1} & γ^{k_{2}+1} \\ 1 & 1 & 1 \end{matrix} & | \\ | & \begin{matrix}α^{2} & β^{2} & γ^{2} \\ α & β & γ \\ 1 & 1 & 1 \end{matrix} & | \end{matrix}.

Then c(p^{k}) = c(p^{k}, 1), χ_{π}(p) = αβγ and c̃(p^{k}) = χ_{π}(p)^{−k} c(1, p^{k}). We will also use the notation

[k_{1}, k_{2}] = χ_{u_{0}}(p^{k_{1}+2k_{2}}) c(p^{k_{1}}, p^{k_{2}}).

We sometimes omit the comma if is obvious, so [10] = [1, 0]. Each coefficient [k_{1}, k_{2}] is the character of an irreducible representation of GL(3, ℂ) evaluated at the semisimple conjugacy class with eigenvalues α′ = χ_{u_{0}}(p)α, β′ = χ_{u_{0}}(p)β, γ′ = χ_{u_{0}}(p)γ. We define [n, m] = 0 if either n or m is negative.

If λ = 1 or p, we want to compute a_{λu_{0},p^{k}}^{(θ)}, where 0 ≤ k ≤ 6θ. The formulas are simpler when λ = p. The general formula in this case is:

(2.1) a_{pu_{0},p^{2k}}^{(θ)} = ∑_{r≥0} Np^{k+r}( ∑_{\begin{matrix}r_{1},r_{2}≥0 \\ r_{1}+2r_{2}≤k−3r \\ 2r_{1}+r_{2}≤3θ−k−3r \\ r_{1}−r_{2}≡k(mod3) \end{matrix}} χ_{π}(p)^{(2/3)(k−r_{1}−2r_{2})}[2r_{1}, 2r_{2}]).

Though the sum over r in (2.1) is infinite, almost all terms are zero since the inner summation conditions are empty for r sufficiently large. The odd coefficients a_{pu_{0},p^{2k+1}}^{(θ)} vanish as noted in Section 2. From formula (2.1) it is not difficult to check the functional equation (1.8). One may verify the other relations such as (R1–R6) in Theorem 1.2 using these formulas and Pieri’s formula in the form

[t, 0][r_{1}, r_{2}] = ∑_{\begin{matrix}m,n≥0 \\ n≤r_{1} \\ 0≤t−m−n≤r_{2} \end{matrix}}[r_{1} + m − n, r_{2} + 2n + m − t]

and its special case [r_{1}, r_{2}] = [r_{1}, 0][0, r_{2}] − [r_{1} − 1, 0][0, r_{2} − 1].

SUMS OF TWISTED GL(3) AUTOMORPHIC L-FUNCTIONS 17

The formula for a_{u_{0},p^{2k}}^{(θ)} is given by:

(2.2) a_{u_{0},p^{2k}}^{(θ)} = ∑_{r=1}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r+6 \\ 2r_{1}+r_{2}≤6θ−2k−6r+6 \\ r_{1}≡r_{2}≡0 (mod 2) \\ r_{1}−r_{2}≡2k (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=0}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r \\ 2r_{1}+r_{2}≤6θ−2k−6r−3 \\ r_{1}≡0,r_{2}≡1 (mod 2) \\ r_{1}−r_{2}≡2k (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=0}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r−3 \\ 2r_{1}+r_{2}≤6θ−2k−6r \\ r_{1}≡1,r_{2}≡0 (mod 2) \\ r_{1}−r_{2}≡2k (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=0}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r−3 \\ 2r_{1}+r_{2}≤6θ−2k−6r−3 \\ r_{1}≡r_{2}≡1 (mod 2) \\ r_{1}−r_{2}≡2k (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k−r_{1}−2r_{2})}[r_{1}, r_{2}].

Once again, though the sums in (2.2) are infinite, almost all terms are zero since each of the four inner summation conditions is empty for r sufficiently large. The formula for a_{u_{0},p^{2k+1}}^{(θ)} is given by:

(2.3) a_{u_{0},p^{2k+1}}^{(θ)} = −( ∑_{r=0}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r−2 \\ 2r_{1}+r_{2}≤6θ−2k−6r−4 \\ r_{1}≡r_{2}≡0 (mod 2) \\ r_{1}−r_{2}≡2k+1 (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k+1−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=1}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r+4 \\ 2r_{1}+r_{2}≤6θ−2k−6r+5 \\ r_{1}≡0,r_{2}≡1 (mod 2) \\ r_{1}−r_{2}≡2k+1 (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k+1−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=1}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r+7 \\ 2r_{1}+r_{2}≤6θ−2k−6r+2 \\ r_{1}≡1,r_{2}≡0 (mod 2) \\ r_{1}−r_{2}≡2k+1 (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k+1−r_{1}−2r_{2})}[r_{1}, r_{2}]

+ ∑_{r=1}^{∞} Np^{k+r−1} ∑_{\begin{matrix}r_{1}+2r_{2}≤2k−6r+7 \\ 2r_{1}+r_{2}≤6θ−2k−6r+5 \\ r_{1}≡r_{2}≡1 (mod 2) \\ r_{1}−r_{2}≡2k+1 (mod 3) \end{matrix}} χ_{π}(p)^{(1/3)(2k+1−r_{1}−2r_{2})}[r_{1}, r_{2}]).

The formulas for the b’s are simpler than those for the a’s. We have, for k even,

(2.4) b_{p,p^{k}}^{(θ)} = ∑_{j=−k/2}^{min(k/2,2θ−3k/2)} ∑_{\begin{matrix}0≤r_{2}≤min( \frac{k}{2} −|j|,θ− \frac{k}{2} −|k+j−θ|) \\ r_{2}≡j−k/2 (mod 2) \end{matrix}} Np^{k+j}×

\frac{1}{2} min (\frac{k}{2} − |j| − r_{2} + 2, θ − \frac{k}{2} − |k + j − θ| − r_{2} + 2, −\frac{3k}{2} − 3j + 2θ + 2 − r_{2}) ×

χ_{π}(p)^{(k/2)+j−r_{2}} [2θ − \frac{3k}{2} − 3j − r_{2} + 1, 2r_{2}] ,

while b_{p,p^{k}}^{(θ)} = 0 if k is odd. Also if k is even:

(2.5) b_{1,p^{k}}^{(θ)} = ∑_{j=−k/2}^{min(k/2,2θ−3k/2)} ∑_{\begin{matrix}0≤r_{2}≤min( \frac{k}{2} −|j|,θ− \frac{k}{2} −|k+j−θ|) \\ r_{2}≡j−k/2 (mod 2) \end{matrix}} Np^{k+j}×

\frac{1}{2} min (\frac{k}{2} − |j| − r_{2} + 2, θ − \frac{k}{2} − |k + j − θ| − r_{2} + 2, −\frac{3k}{2} − 3j + 2θ + 2 − r_{2}) ×

χ_{π}(p)^{(k/2)+j−r_{2}} [2θ − \frac{3k}{2} − 3j − r_{2}, 2r_{2}] .

If k is odd:

(2.6) b_{1,p^{k}}^{(θ)} =

− ∑_{j=− \frac{k−1}{2}}^{min(\frac{k−1}{2} ,2θ− \frac{3k+1}{2})} ∑_{\begin{matrix}0≤r_{2}≤min( \frac{k−1}{2} −|j|,θ− \frac{k+1}{2} −|k+j−θ|) \\ r_{2}≡j−(k−1)/2 (mod 2) \end{matrix}} Np^{k−1+j}×

\frac{1}{2} min (\frac{k − 1}{2} −|j|−r_{2} +2, θ − \frac{k + 1}{2} −|k +j −θ|−r_{2} +2, 2θ − \frac{3(k − 1)}{2} −3j −r_{2})×

χ_{π}(p)^{( \frac{k−1}{2} )+j−r_{2}} [2θ − \frac{3(k − 1)}{2} − 3j − r_{2}, 2r_{2}] .

Next we offer some remarks about the stable a-coefficients. Let F (k) be the k-th symmetric power of the symmetric square character [2, 0]. This is a homogeneous polynomial of degree 2k in α′, β′ and γ′ given by the formula

F (k) = ∑_{\begin{matrix}r_{1}+2r_{2}≤k \\ r_{1}+2r_{2}≡k (mod 3) \end{matrix}}[2r_{1}, 2r_{2}].

Define F (k) = 0 if k < 0. Using (S1)–(S3) the stable coefficients may be defined by a_{λ,p^{k}}^{stab} = a_{λ,p^{k}}^{(θ)} whenever θ ≥ k/2. Then one may check that:

(2.7) a_{u_{0}p,p^{2k}}^{stab} = Np^{k} F (k) + Np^{k+1} χ_{π}(p)^{2} F (k − 3) + Np^{k+2} χ_{π}(p)^{4} F (k − 6) + ⋯
a_{u_{0},p^{2k}}^{stab} = a_{u_{0}p,p^{2k}}^{stab} + [0, 1]a_{u_{0}p,p^{2k−2}}^{stab},

a_{u_{0},p^{2k+1}}^{stab} = −([1, 0]a_{u_{0},p^{2k}}^{stab} + χ_{π}(p) a_{u_{0}p,p^{2k−2}}^{stab}).

Introducing a topic to be developed further in the next section, we observe now that the symmetric square L-function appears as a residue of the double Dirichlet series, and one can see its functional equation this way. Indeed, the group of functional equations is generated by

(s, w) ↦ (s + w − \frac{1}{2}, 1 − w), (s, w) ↦ (1 − s, w + 3s − \frac{3}{2}).

One element of this dihedral group of order twelve is:

(s, w) ↦ (\frac{3}{2} − s − w, w),

so when taking the residue at w = 1, we obtain a Dirichlet series having a functional equation under s → \frac{1}{2} − s. Now

(2.8)  ∑_{t=0}^{2θ} Np^{−t}b_{1,p^{t}}^{(θ)} = F (θ) + Np F (θ − 3) + Np^{2}F (θ − 6) + ….

The significance of (2.8) is that this Dirichlet series is

L(2s, f, ∨^{2}) ζ(6s − 1)

This follows from (2.7) since

Np^{θ} ∑_{t=0}^{2θ} Np^{−t}b_{1,p^{t}}^{(θ)} = a_{p,p^{2θ}}^{stab}.

We have also investigated the local matters reported on here for r = 4. Globally and locally this is a fundamentally different situation, for in that case the group of functional equations is infinite. We find that there the local combinatorial problem, discussed above for r = 3, does have a solution which is almost but not quite unique. In the Dirichlet polynomial P the middle coefficient a_{1,p^{4α}}^{(α)} is not determined and can be chosen freely—there will still be a solution to the combinatorial problem of Section 1. The residue will be the symmetric square times a Dirichlet series with a natural boundary. Nevertheless computer computations suggest that there might be a choice which makes this Dirichlet series as simple as possible. Let [n_{1}, n_{2}, n_{3}] denote the L-group character with highest weight ∑ n_{i}ϖ_{i} in terms of the fundamental dominant weights ϖ_{i}. Assuming for simplicity that the central character χ_{π} is trivial, this preferred choice would seem to be:

a_{1,p^{4}}^{(1)} = [0, 2, 0]Np^{2} + [1, 0, 1]Np + 1,

a_{1,p^{8}}^{(2)} =

Np^{4}([2, 0, 2]+[0, 4, 0]+1)+Np^{3}([0, 1, 2]+[1, 0, 1]+[1, 2, 1]+[2, 1, 0])+Np^{2}[0, 2, 0].

If we make these choices then the residue is a series

∑_{α} Z[α] Np^{−2αs},

where in terms of the symmetric square [2, 0, 0] we have:

Z[1] = [2, 0, 0],
Z[2] = ∨^{2}([2, 0, 0]),
Z[3] = ∨^{3}([2, 0, 0]) + [0, 0, 2] Np,
Z[4] = ∨^{4}([2, 0, 0]) + [2, 0, 2] Np + Np^{2},
Z[5] = ∨^{5}([2, 0, 0]) + ([4, 0, 2] + [0, 2, 2] − [0, 1, 0]) Np + 2[2, 0, 0] Np^{2}.

For different choices of coefficients these formulae would be considerably more complicated.

3. An application to automorphic forms on GL(3). In this section our objective will be to illustrate the technique of multiple Dirichlet series in the particular case of automorphic L-series on GL(3) defined over ℚ. We change slightly the notation from Sections 0–2 in order to state our results in a way which closely resembles the classical GL(2) theory.

Let f be an automorphic cusp form on GL(3) over ℚ of level N corresponding to the automorphic representation π. All L-series in this section will have the archimedean factor removed (we shall deal with this factor explicitly and separately below). Assume that f is an eigenfunction of all the Hecke operators. Let S be a finite set of primes including 2 and the primes dividing N and let M = ∏_{p∈S} p. As remarked before, the L-series associated to f has the form

L(s, f) = ∑_{1}^{∞} \frac{c(m)}{m^{s}}

for certain generalized Fourier coefficients c(m), and decomposes into the Euler product

(3.1)  L(s, f) = ∏_{p} (1 − α_{p}p^{−s})^{−1}(1 − β_{p}p^{−s})^{−1}(1 − γ_{p}p^{−s})^{−1},

the product being over all primes p of ℚ.

It is well known that L(s, f) possesses a functional equation:

(3.2)  N^{s/2}G_{f}(s)L(s, f) = ϵ_{f}N^{(1−s)/2}G_{f}(1 − s)L(1 − s, f̃),

where f̃ denotes the contragredient of f and G_{f} is a product of gamma factors depending on f.

If f is twisted by a character χ of conductor D the associated L-series becomes

(3.3) L(s, f, χ) = ∏_{p} (1 − χ(p)α_{p}p^{−s})^{−1}(1 − χ(p)β_{p}p^{−s})^{−1}(1 − χ(p)γ_{p}p^{−s})^{−1},

and the functional equation is given by

(3.4) (D^{3}N)^{s/2}G_{f,χ}(s)L(s, f, χ)

= ϵ_{f}τ(χ)^{3}ψ_{N}(D)(D^{3}N)^{(1−s)/2}G_{f,χ}(1 − s)L(1 − s, f̃, \overline{χ}).

Here the gamma factors have an additive shift that depends on the sign of χ(−1), ψ_{N} is the Dirichlet character associated to the idèle class character χ_{π}, and τ(χ) is the Gauss sum associated to χ normalized to have absolute value equal to 1. This is simply (0.4) restated in classical language. In this section we will also assume that ψ_{N} is quadratic or trivial of conductor dividing N (this assumption simplifies our work but is not essential for it).

In what follows we will not need a very explicit description of G_{f,χ} but we will find the following upper bound convenient. For σ_{1} > σ_{2} and t real it follows from Stirling’s formula that for large |t|, independent of χ,

(3.5)  |G_{f,χ}(σ_{1} + it)/G_{f,χ}(σ_{2} − it)| ≪ (|t| + 1)^{3(σ_{1}−σ_{2})/2}.

Here the implied constant depends only on the eigenvalues of f.

When all finite primes are included in the product (3.1) the functional equation (3.2) has its optimal form. However, it is often convenient to omit factors corresponding to “bad” primes, for example 2 and those dividing the level N. For M, S as above we denote the L-series with Euler factors corresponding to primes dividing M removed as follows:

(3.6) L_{M}(s, f, χ) = ∏_{p∉S} (1 − χ(p)α_{p}p^{−s})^{−1}(1 − χ(p)β_{p}p^{−s})^{−1}(1 − χ(p)γ_{p}p^{−s})^{−1}

= L_{M}(s, f, χ) ∏_{p∈S} (1 − χ(p)α_{p}p^{−s})(1 − χ(p)β_{p}p^{−s})(1 − χ(p)γ_{p}p^{−s}).

A particularly interesting class of twists are the quadratic characters described by χ_{d}(∗) = (\frac{d}{∗}), where d is a square-free non-zero integer and χ_{d} is the real character associated to the quadratic field ℚ(\sqrt{d}). We will assume in the following that the character ω that we twist by is quadratic.

When twisted by χ_{d}, the L-series L(s, f, χ_{d}) will have a functional equation of the form (3.4) when χ_{d} is a primitive character. This corresponds to the case where d is the square-free part of a fundamental discriminant. In the preceding sections, it was noted that when d is not square-free, it is possible to complete L(s, f, χ_{d}) by multiplying by a certain Dirichlet polynomial in such a way that the resulting product has a functional equation of precisely the same form (3.4), with D replaced by |d| or |4d|.

Our object will be to obtain the analytic continuation in s, w, and an estimate for the growth in vertical strips w = σ +it, for fixed σ and s, of the following double Dirichlet series. Let l_{1}, l_{2} > 0, l_{1}, l_{2}|M and a_{1}, a_{2} ∈ {1, −1} and let χ_{a_{1}l_{1}}, χ_{a_{2}l_{2}} be the quadratic characters corresponding to a_{1}l_{1}, a_{2}l_{2} as defined above. For l_{1}, l_{2} > 0, l_{1}, l_{2}|M and a_{1}, a_{2} ∈ {1, −1} define

(3.7) Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) = ∑_{(d,M)=1} \frac{L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}})χ_{a_{2}l_{2}}(d_{0})P_{a_{1}l_{1}d_{0},d_{1}}(s)}{d^{w}},

where we sum over d > 0 and use the decomposition d = d_{0}d_{1}^{2}, with d_{0} > 0 squarefree. In the notation of Section 1, Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) = Z(s, w; π, ω). Here π corresponds to f ⊗ χ_{a_{1}l_{1}} and χ_{a_{2}l_{2}} replaces ω.

Proceeding as in [DGH], the following proposition will provide a useful way of collecting the properties of Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}). For a positive integer M, define

Div(M) = {a · l | a = ±1, 1 ≤ l, l|M},

which has cardinality 2d(M) = 2 ∑_{d|M} 1. Let \overrightarrow{Z}_{M,f}(s, w; χ_{a_{2}l_{2}}, χ_{Div(M)}) denote the 2d(M) by 1 column vector whose j^{th} entry is Z_{M}(s, w; χ_{a_{2}l_{2}}, χ^{(j)}), where χ^{(j)} (j = 1, 2, …, 2d(M)) ranges over the characters χ_{a_{1}l_{1}} with a_{1} = ±1, 1 ≤ l_{1}, l_{1}|M. Then, we will prove

Proposition 3.1. There exists a 2d(M) by 2d(M) matrix Φ^{(a_{2}l_{2})}(w) such that for any fixed w, w ≠ 1, and for any s with sufficiently large real part (depending on w)

∏_{p|(M/l_{2})} (1 − p^{−2+2w}) · \overrightarrow{Z}_{M,f}(s, w; χ_{a_{2}l_{2}}, χ_{Div(M)})

= Φ^{(a_{2}l_{2})}(w)\overrightarrow{Z}_{M,f}(s + w − 1/2, 1 − w; χ_{a_{2}l_{2}}, χ_{Div(M)}).

The entries of Φ^{(a_{2}l_{2})}(w), denoted by Φ_{i,j}^{(a_{2}l_{2})}(w), are meromorphic functions in ℂ. Also, the only possible poles of the Φ_{i,j}^{(a_{2}l_{2})}(w) are canceled by trivial zeros of the L-series appearing in the numerator of \overrightarrow{Z}_{M,f}(s + w − 1/2, 1 − w; χ_{a_{2}l_{2}}, χ_{Div(M)}).

Proof: By the basic identity (1.1), (1.2), writing n = n_{0}n_{1}^{2}, with square-free n_{0} > 0, we have (3.8)

Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) = ∑_{(n,M)=1} \frac{L_{M}(w, χ̃_{n_{0}}χ_{a_{2}l_{2}})χ_{a_{1}l_{1}}(n_{0})c(n_{0}n_{1}^{2})Q_{n_{0},n_{1}}^{(a_{2}l_{2})}(w)}{n^{s}}.

Here χ̃_{n_{0}} denotes the quadratic character (∗/n_{0}). Now

(3.9)  L_{M}(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) = L(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) · ∏_{p|M} (1 − χ̃_{n_{0}}χ_{a_{2}l_{2}}(p)p^{−w}),

where L(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) satisfies the functional equation

(3.10) G_{ϵ}(w)(n_{0}l_{2}D_{a_{2}l_{2}})^{w/2}L(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) =

G_{ϵ}(1 − w)(n_{0}l_{2}D_{a_{2}l_{2}})^{(1−w)/2}L(1 − w, χ̃_{n_{0}}χ_{a_{2}l_{2}}).

Here ϵ = χ̃_{n_{0}}χ_{a_{2}l_{2}}(−1),

(3.11)  G_{ϵ}(w) = \{ \begin{matrix}π^{−w/2}Γ(w/2) & if ϵ = 1 \\ π^{−(w+1)/2}Γ((w + 1)/2) & if ϵ = −1, \end{matrix}

and

(3.12)  D_{a_{2}l_{2}} = \{ \begin{matrix}1 & if a_{2}l_{2} ≡ 1 (mod 4) \\ 4 & otherwise. \end{matrix}

Combining this with the functional equation for Q implied by (0.8), we obtain

Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})

= ∑_{a_{3}=1,−1} ∑_{(n,M)=1, n≡a_{3} (4)} \frac{G_{ϵ(a_{3}a_{2}l_{2})}(1 − w)(l_{2}D_{a_{2}l_{2}})^{1/2−w}}{G_{ϵ(a_{3}a_{2}l_{2})}(w)n^{s+w−1/2}}

×χ_{a_{1}l_{1}}(n_{0})L_{M}(1−w, χ̃_{n_{0}}χ_{a_{2}l_{2}})c(n_{0}n_{1}^{2})Q_{n_{0},n_{1}}^{(a_{2}l_{2})}(1−w)· ∏_{p|(M/l_{2})} (1−χ̃_{n_{0}}χ_{a_{2}l_{2}}(p)p^{−w})

× ∏_{p|(M/l_{2})} (1 − χ̃_{n_{0}}χ_{a_{2}l_{2}}(p)p^{−1+w})^{−1}.

Here ϵ(a) denotes the sign of a. Note that we are leaving out terms in the product where p|l_{2} as the character vanishes here.

Multiplying by ∏_{p|(M/l_{2})} (1 − p^{−2+2w}), reorganizing, and proceeding exactly as in the proof of Proposition 4.2 in [DGH] we obtain (in the case of a_{2}l_{2} ≡ 1 (mod 4))

(3.13) ∏_{p|(M/l_{2})} (1 − p^{−2+2w}) · Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})

= \frac{1}{2} · l_{2}^{1/2−w} · ∑_{l_{3},l_{4}|(M/l_{2})} μ(l_{3})χ_{a_{2}l_{2}}(l_{3}l_{4})l_{3}^{−w}l_{4}^{−1+w} ∑_{a_{3}=1,−1} \frac{G_{ϵ(a_{3}a_{2}l_{2})}(1 − w)}{G_{ϵ(a_{3}a_{2}l_{2})}(w)}

× (Z_{M}(s + w − 1/2, 1 − w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}l_{3}l_{4}})+

a_{3}Z_{M}(s + w − 1/2, 1 − w; χ_{a_{2}l_{2}}, χ_{−a_{1}l_{1}l_{3}l_{4}})).

In the case a_{2}l_{2} ≡ −1, 2 (mod 4), just the behavior at the finite place 2 changes. The observation concerning the poles of Φ_{i,j}^{(a_{2}l_{2})}(w) follows from the fact that the only source of possible poles on the right hand side of (3.13) are the gamma factors G_{ϵ(a_{3}a_{2}l_{2})}(1 − w). These, however, are canceled by trivial zeros of L-series.

This completes the proof of Proposition 3.1.

The function Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) defined in (3.7) also possesses a functional equation as s → 1 − s. Let d(M) be as before, and let \overrightarrow{Z}_{M,f}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}}) denote the 2d(M) by 1 column vector whose j^{th} entry is Z_{M}(s, w; χ^{(j)}, χ_{a_{1}l_{1}}), where χ^{(j)} (j = 1, 2, …, 2d(M)) ranges over the characters χ_{a_{2}l_{2}} with a_{2} = ±1, 1 ≤ l_{2}, l_{2}|M.

Then we have the following.

Proposition 3.2. There exists a 2d(M) by 2d(M) matrix Ψ^{(a_{1}l_{1})}(s) such that for any fixed s, s ≠ 1, and for any w with sufficiently large real part (depending on s)

\overrightarrow{Z}_{M,f}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}}) · ∏_{p|(M/l_{1})} (1 − α̃_{p}^{2}p^{−2+2s})(1 − β̃_{p}^{2}p^{−2+2s})(1 − γ̃_{p}^{2}p^{−2+2s})

= Ψ^{(a_{1}l_{1})}(s)\overrightarrow{Z}_{M,f̃}(1 − s, w + 3s − 3/2; χ_{Div(M)}, χ_{a_{1}l_{1}}).

The entries of Ψ^{(a_{1}l_{1})}(s), denoted by Ψ_{i,j}^{(a_{1}l_{1})}(s), are meromorphic functions in ℂ. The only possible poles of Ψ_{i,j}^{(a_{1}l_{1})}(s) are canceled by trivial zeros of the L-series occurring in the numerator of \overrightarrow{Z}_{M,f̃}(1 − s, w + 3s − 3/2; χ_{Div(M)}, χ_{a_{1}l_{1}}).

Proof: First, write

(3.14) L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}}) = L(s, f, χ_{d_{0}}χ_{a_{1}l_{1}})

· ∏_{p|(M/l_{1})} (1 − α_{p}χ_{d_{0}}χ_{a_{1}l_{1}}(p)p^{−s})(1 − β_{p}χ_{d_{0}}χ_{a_{1}l_{1}}(p)p^{−s})(1 − γ_{p}χ_{d_{0}}χ_{a_{1}l_{1}}(p)p^{−s})

= L(s, f, χ_{d_{0}}χ_{a_{1}l_{1}})

· ∑_{l_{α}|(M/l_{1})} μ(l_{α})χ_{a_{1}d_{0}l_{1}}(l_{α})l_{α}^{−s} ∏_{p|l_{α}} α_{p}

· ∑_{l_{β}|(M/l_{1})} μ(l_{β})χ_{a_{1}d_{0}l_{1}}(l_{β})l_{β}^{−s} ∏_{p|l_{β}} β_{p} ∑_{l_{γ}|(M/l_{1})} μ(l_{γ})χ_{a_{1}d_{0}l_{1}}(l_{γ})l_{γ}^{−s} ∏_{p|l_{γ}} γ_{p}.

By (3.4)

(3.15) L(s, f, χ_{d_{0}}χ_{a_{1}l_{1}})

= ϵ_{f}ψ_{N}(a_{1}d_{0}l_{1})((d_{0}l_{1}D_{a_{1}d_{0}l_{1}})^{3}N)^{(1/2−s)} \frac{G_{f,ϵ}(1 − s)}{G_{f,ϵ}(s)}L(1 − s, f̃, χ_{d_{0}}χ_{a_{1}l_{1}}),

where G_{f,ϵ} is a product of gamma factors depending only on f and on the sign of a_{1}d_{0}l_{1}, and D_{a_{1}d_{0}l_{1}} is given by (3.12).

On the other side of the functional equation,

L(1 − s, f̃, χ_{a_{1}d_{0}l_{1}}) = L_{M}(1 − s, f̃, χ_{a_{1}d_{0}l_{1}}) ∏_{p|(M/l_{1})}

(1 − α̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})^{−1}(1 − β̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})^{−1}(1 − γ̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})^{−1},

so

L(1 − s, f̃, χ_{a_{1}d_{0}l_{1}}) ∏_{p|(M/l_{1})} (1 − α̃_{p}^{2}p^{−2+2s})(1 − β̃_{p}^{2}p^{−2+2s})(1 − γ̃_{p}^{2}p^{−2+2s})

= L_{M}(1 − s, f̃, χ_{a_{1}d_{0}l_{1}})

· ∏_{p|(M/l_{1})} (1 + α̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})(1 + β̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})(1 + γ̃_{p}χ_{a_{1}d_{0}l_{1}}(p)p^{−1+s})

= L_{M}(1 − s, f̃, χ_{a_{1}d_{0}l_{1}}) ∑_{lα̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lα̃)l−1+s ; α̃ ∏_{p|lα̃} α̃_{p}

· ∑_{lβ̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lβ̃)l−1+s ; β̃ ∏_{p|lβ̃} β̃_{p} ∑_{lγ̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lγ̃)l−1+s ; γ̃ ∏_{p|lγ̃} γ̃_{p}.

Combining the above with (3.14) and (3.15) we obtain

(3.16) Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) ∏_{p|(M/l_{1})} (1 − α̃_{p}^{2}p^{−2+2s})(1 − β̃_{p}^{2}p^{−2+2s})(1 − γ̃_{p}^{2}p^{−2+2s})

= ∑_{(d,M)=1} ϵ_{f}ψ_{N}(a_{1}d_{0}l_{1})((l_{1}D_{a_{1}d_{0}l_{1}})^{3}N)^{(1/2−s)} \frac{G_{f,ϵ}(1 − s)}{G_{f,ϵ}(s)} · \frac{L_{M}(1 − s, f̃, χ_{d_{0}a_{1}l_{1}})}{d^{w+3s−3/2}}

· P_{d_{0},d_{1}}^{(a_{1}l_{1})}(1 − s)χ_{a_{2}l_{2}}(d_{0}) ∑_{l_{α}|(M/l_{1})} μ(l_{α})χ_{a_{1}d_{0}l_{1}}(l_{α})l_{α}^{−s} ∏_{p|l_{α}} α_{p}

· ∑_{l_{β}|(M/l_{1})} μ(l_{β})χ_{a_{1}d_{0}l_{1}}(l_{β})l_{β}^{−s} ∏_{p|l_{β}} β_{p} ∑_{l_{γ}|(M/l_{1})} μ(l_{γ})χ_{a_{1}d_{0}l_{1}}(l_{γ})l_{γ}^{−s} ∏_{p|l_{γ}} γ_{p}

· ∑_{lα̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lα̃)l−1+s ; α̃ ∏_{p|lα̃} α̃_{p}

· ∑_{lβ̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lβ̃)l−1+s ; β̃ ∏_{p|lβ̃} β̃_{p} ∑_{lγ̃|(M/l_{1})} χ_{a_{1}d_{0}l_{1}}(lγ̃)l−1+s ; γ̃ ∏_{p|lγ̃} γ̃_{p}.

This decomposes into a linear combination of the functions Z̃_{M}(1 − s, w + 3s − 3/2; χ^{∗}, χ_{a_{1}l_{1}}) where the character χ^{∗} takes one of the two forms

ψ_{N}χ_{l_{α}l_{β}l_{γ}lα̃lβ̃lγ̃} χ_{a_{2}l_{2}}, ψ_{N}χ_{−1}χ_{l_{α}l_{β}l_{γ}lα̃lβ̃lγ̃} χ_{a_{2}l_{2}}.

Here we have denoted by Z̃_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}), the double Dirichlet series formed with f̃ rather than f. The result then follows as in the previous proposition and as in [DGH] and we omit the precise, rather complicated, expression for Ψ^{(a_{1}l_{1})}(s). This completes the proof of Proposition 3.2.

We now continue the analytic continuation of Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}). Our approach will follow a parallel route to that of [DGH], differing only in some details caused by the fact that the Ramanujan conjecture is not known for automorphic forms on GL(3). We refer to [DGH] for background and definitions that will be used from the theory of several complex variables, and for some details of arguments that we will omit below.

We will repeatedly apply the functional equations given in Propositions 3.1, 3.2. To do this conveniently, we define two involutions on ℂ × ℂ :

α : (s, w) → (1 − s, w + 3s − 3/2) and β : (s, w) → (s + w − 1/2, 1 − w).

Then α, β generate D_{12}, the dihedral group of order 12, and α^{2} = β^{2} = 1, (αβ)^{6} = (βα)^{6} = 1. Note that αβ ≠ βα.

We will find it useful in the following to define three regions R_{1}, R_{2}, R_{3} as follows: Write s, w as s = σ + it, w = ν + iγ.

The tube region R_{1} is defined to be the set of all points (s, w) such that (σ, ν) lie strictly above the polygon determined by (−2/5, 37/10), (3/2, 0), and the rays ν = −3σ + 5/2 for σ ≤ −2/5 and ν = −σ + 3/2 for σ ≥ 3/2.

The tube region R_{2} is defined to be the set of all points (s, w) such that (σ, ν) lie strictly above the polygon determined by (−1/2, 3) and (3/2, 0) and the rays ν = −2σ + 2 for σ ≤ −1/2, and ν = −σ + 3/2 for σ ≥ 3/2.

The tube region R_{3} is defined to be the set of all points (s, w) such that (σ, ν) lie strictly above the line ν = −2σ + 2.

These regions are related by the involutions α, β as described in the following proposition. The proof, a simple exercise, is omitted.

Proposition 3.3. The regions R_{1} and α(R_{1}) have a non-empty intersection, and the convex hull of R_{1} ∪ α(R_{1}) equals R_{2}. Similarly, R_{2} and β(R_{2}) have a non-empty intersection and the convex hull of R_{2} ∪ β(R_{2}) equals R_{3}. Finally, R_{3} and α(R_{3}) have a non-empty intersection and the convex hull of R_{3} ∪ α(R_{3}) equals ℂ^{2}.

We will begin by demonstrating

Proposition 3.4. Let R_{1} be the tube region defined above. The functions

(w − 1)Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) and (w − 1)Z̃_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})

are analytic in R_{1}.

Proof: Consider first the expression for Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) given in (3.7). We first need to determine when L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}}) converges absolutely. A well known upper bound for the size of the coefficients c(n) of f (and c̃(n) of f̃ ) is c(n) ≪ |n|^{2/5+ϵ}. This follows from the properties of the Rankin-Selberg convolution of f with f̃ (in particular from the presence of nine gamma factors in the functional equation). As a consequence, L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}}) converges absolutely for σ > 7/5 and is bounded above by a constant independent of d_{0}. See Jacquet, Piatetski-Shapiro and Shalika [JPS1] and Jacquet and Shalika [JS2] for the Rankin-Selberg method for GL(n). Over a number field a weaker estimate could instead be obtained from Jacquet, Piatetski-Shapiro and Shalika [JPS2].

Now suppose that the sum were restricted only to square-free d = d_{0}. It is clear then that Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) would converge absolutely in the region σ > 7/5, ν > 1. Reflecting s to 1 − s (using (3.4)) and noting that the missing Euler factors do not affect convergence, one sees that there would be convergence in the region σ < −2/5, ν > −3σ + 5/2 and consequently, by the Phragmen-Lindelöf principle, there would be convergence in the region lying between these two areas and above the line connecting (−2/5, 37/10) and (7/5, 1). It follows from (2.1–2.3) that we have the bound

P_{a_{1}l_{1}d_{0},d_{1}}(s) ≪ 1

for σ > 7/5. In fact, more precisely, P_{a_{1}l_{1}d_{0},d_{1}}(s) ≪ 1 will hold whenever σ > 1/2 + η, where c(m) ≪_{ϵ} m^{η+ϵ} is an upper bound for the coefficients of the GL(3) form f. Thus, for example, if f is the adjoint square lift of a form on GL(2) then η = 2/5 will produce the estimate above.

Because of this, and the functional equation (0.5) applied to P_{a_{1}l_{1}d_{0},d_{1}}(s), precisely the same estimates apply as we sum over all d. Thus Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) converges above the given lines. As f is cuspidal L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}}) has no poles in this region.

Noting that the expression converges when ν > 1, σ > 7/5, we now change the order of summation to the form given in (3.8). The location of the pole of the Rankin-Selberg convolution of f with its contragredient implies the absolute convergence of L_{M}(s, f) for σ > 1. Consequently, applying the usual convexity estimates for L(w, χ_{n_{0}}), the corresponding estimate

Q_{n_{0},n_{1}}^{(a_{2}l_{2})}(w) ≪ c(n_{0}n_{1}^{2})

(which follows from (2.4–2.6)) and functional equations for c(n_{0}n_{1}^{2})Q_{n_{0},n_{1}}^{(a_{2}l_{2})}(w), we see Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) converges for σ > 1 when ν > 1, for σ > (−1/2)ν + 3/2 when 0 ≤ ν ≤ 1 and σ > −ν + 3/2 when ν < 0. The factor w − 1 cancels the pole at w = 1. The regions described above overlap and thus by the convexity principle for several complex variables (see Proposition 4.6 of [DGH]) Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})P (s, w) has an analytic continuation to the convex closure of the regions, which is R_{1} described above. An identical argument applies when f is replaced by f̃.

This completes the proof of Proposition 3.4.

Our plan is now to apply the involutions α, β, α in that order to R_{1}, and use Propositions 3.1 and 3.2 to extend the analytic continuation to ℂ^{2}. To aid in this, it will be useful to introduce some additional notation to make the content of these propositions a bit clearer and easier to apply. Let

A(s, w) ≡ A_{M}(s, w) = ∏_{p|M} (1 − α̃_{p}^{2}p^{−2+2s})(1 − β̃_{p}^{2}p^{−2+2s})(1 − γ̃_{p}^{2}p^{−2+2s})

and

(3.17)  B(s, w) ≡ B_{M}(s, w) = ∏_{p|M}(1 − p^{−2+2w}),

and let

Ψ̃^{(a_{1}l_{1})}(s, w) = Ψ^{(a_{1}l_{1})}(s) ∏_{p|l_{1}} (1 − α̃_{p}^{2}p^{−2+2s})(1 − β̃_{p}^{2}p^{−2+2s})(1 − γ̃_{p}^{2}p^{−2+2s}),

Φ̃^{(a_{2}l_{2})}(s, w) = Φ^{(a_{2}l_{2})}(w) ∏_{p|l_{2}}(1 − p^{−2+2w}).

The following is a reformulation of the content we require now from Propositions 3.1 and 3.2. For (s, w) such that both sides are contained in a connected region of analytic continuation for (w − 1)Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})

(3.18)

A(s, w)\overrightarrow{Z}_{M,f}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}}) = Ψ̃^{(a_{1}l_{1})}(s, w)\overrightarrow{Z}_{M,f̃}(α(s, w); χ_{Div(M)}, χ_{a_{1}l_{1}})

and (3.19) B(s, w)\overrightarrow{Z}_{M,f}(s, w; χ_{a_{2}l_{2}}, χ_{Div(M)}) = Φ̃^{(a_{2}l_{2})}(s, w)\overrightarrow{Z}_{M,f̃}(β(s, w); χ_{a_{2}l_{2}}, χ_{Div(M)}).

Next we prove the analytic continuation of Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}).

Proposition 3.5. Let

P(s, w) = w(w − 1)(3s + w − 5/2)(3s + 2w − 3)(3s + w − 3/2).

Then the following product has an analytic continuation to an entire function in ℂ^{2} :

Z^{∗}_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) :=

A(s, w)A(α(s, w))A(β(s, w))A(βα(s, w))B(s, w)B(α(s, w))P(s, w)

× Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}).

Proof: Let P (s, w) = w − 1. In Proposition 3.4 we established the analytic continuation of Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})P (s, w) and Z̃_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})P (s, w) in R_{1}. As α^{2} = 1, ˜ f̃ = f and Ψ̃^{(a_{1}l_{1})}(s, w) is meromorphic in ℂ^{2}, it follows that

Ψ̃^{(a_{1}l_{1})}(s, w)\overrightarrow{Z}_{M,f}(α(s, w); χ_{Div(M)}, χ_{a_{1}l_{1}})P (α(s, w))

is a meromorphic function in α(R_{1}). As any poles of Ψ̃^{(a_{1}l_{1})}(s, w) arise from gamma factors and are canceled by trivial zeros of the L-series appearing in

\overrightarrow{Z}_{M,f̃}(α(s, w); χ_{Div(M)}, χ_{a_{1}l_{1}}),

we can conclude from Proposition 3.4 and (3.18) that

A(s, w)P (s, w)P (α(s, w))\overrightarrow{Z}_{M,f}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}})

is analytic in R_{1} ∪ α(R_{1}), R_{1} and α(R_{1}) having a substantial intersection (containing ℜ(s), ℜ(w) > 1). Thus by the convexity principle this function is analytic in R_{2}, the convex hull of the union.

A similar argument applies to \overrightarrow{Z}_{M,f̃}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}}). The rest of the argument proceeds exactly as in the proof of Proposition 4.10 of [DGH]. The only change is a slight difference in the definition of the region R_{1}, which does not affect the argument. This completes the proof of Proposition 3.5.

We will now use the analytic continuation and functional equations (3.18), (3.19) for the function \overrightarrow{Z}_{M,f}(s, w; χ_{Div(M)}, χ_{a_{1}l_{1}}) to locate poles and obtain an estimate for the growth of Z_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) in a vertical strip. Before doing this, however, we need some additional notation.

Let \overrightarrow{Z}_{M,f}(s, w) denote the 4d(M)^{2}–dimensional column vector consisting of the concatenation of the 2d(M) column vectors \overrightarrow{Z}_{M,f}(s, w; χ_{a_{2}l_{2}}, χ_{Div(M)}) for a_{2} ∈ {1, −1} and all l_{2}|M. Then by Propositions 3.1 and 3.2, combined with (3.18), (3.19), there exist 4d(M)^{2} by 4d(M)^{2} matrices Φ_{M}(s, w), Ψ_{M}(s, w) such that

(3.20)  A_{M}(s, w)\overrightarrow{Z}_{M,f}(s, w) = Ψ_{M}(s, w)\overrightarrow{Z}_{M,f̃}(α(s, w))

and

(3.21)  B_{M}(s, w)\overrightarrow{Z}_{M,f}(s, w) = Φ_{M}(s, w)\overrightarrow{Z}_{M,f}(β(s, w)).

Here A_{M}(s, w), B_{M}(s, w) are given by (3.17). The matrices Φ_{M}(s, w), Ψ_{M}(s, w) are constructed from blocks of Φ̃^{(a_{2}l_{2})}(s, w) and Ψ̃^{(a_{1}l_{1})}(s, w) on the diagonal.

We now remark that the function Z^{∗}_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}), defined in Proposition 3.5, is of finite order. This straight forward statement about a one–variable problem requires the theory of several complex variables for the proof. As the proof in this context is virtually identical to that given in detail in Proposition 4.11 of [DGH] we will omit it.

We now show:

Proposition 3.6. Let w = ν+it. For ϵ > 0, −3/4−ϵ ≤ ν, and any a_{1}, a_{2} ∈ {1, −1}, l_{1}, l_{2}|M the function Z_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) is an analytic function of w, except for poles at w = \frac{3}{4}, 1. If (l_{1}, l_{2}) = 1 or 2 and |t| > 1, then it satisfies the upper bounds

Z_{M} (\frac{1}{2}, ν + it; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) ≪_{ϵ} (NM^{3})^{1/4+ϵ},

for 7/4 + ϵ < ν, and

Z_{M} (\frac{1}{2}, ν + it; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) ≪_{ϵ} (NM^{3})^{1/4+ϵ}M^{a(7/4−ν)+v_{1}(ϵ)}|t|^{b(7/4−ν)+v_{2}(ϵ)}

for −\frac{3}{4} − ϵ ≤ ν ≤ \frac{7}{4} + ϵ. The constants a, b are positive and explicitly computable. Also the functions v_{1}(ϵ), v_{2}(ϵ) are explicitly computable functions satisfying

lim_{ϵ→0} v_{1}(ϵ) = lim_{ϵ→0} v_{2}(ϵ) = 0.

Proof: The result stated here is rather crude, but suffices for our purposes. It would be an easy matter refine these estimates but as we are not aiming for an optimal unweighted mean value estimate this is all that we require.

The first bound in the region 7/4 + ϵ < ν follows immediately from the standard convexity estimate L(1/2, f, χ_{D}) ≪ (ND^{3})^{1/4+ϵ}, where N is the level of f and D is the conductor of χ_{D}. The bound for −3/4 − ϵ ≤ ν ≤ 7/4 + ϵ is more difficult to obtain. The approach is to first obtain a bound for Z_{M} (\frac{1}{2}, −3/4 − ϵ + it; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}). We then apply a convexity argument (using the finite order property mentioned above) to complete the proof for −3/4 − ϵ ≤ ν ≤ 7/4 + ϵ. The argument is identical to that presented in the proof of Proposition 4.12 of [DGH] and so will be omitted. We will remark, however, that the only possible poles of Z_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) are at locations corresponding to zeros of the factors A, B and P appearing on the right hand side of the product given in Proposition 3.5. All can be eliminated as possibilities except for w = 1 and w = 3/4. This completes the proof of Proposition 3.6.

It now remains to calculate the order of the pole of Z_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) and compute the leading coefficient in the Laurent expansion at w = 1.

To do this we first compute the residue of Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) at w = 1 for s in a neighborhood of 1/2. Our result is summarized in

Proposition 3.7. For s in a neighborhood of 1/2 but s ≠ 1/2, the function Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) is analytic at w = 1 whenever a_{2}l_{2} ≠ 1. When a_{2}l_{2} = 1, Z_{M}(s, w; 1, χ_{a_{1}l_{1}}) has a pole of order 1 at w = 1 with residue

R_{M}(s; χ_{a_{1}l_{1}}) = lim_{w→1}(w − 1)Z_{M}(s, w; 1, χ_{a_{1}l_{1}})

given by:

R_{M}(s; χ_{a_{1}l_{1}}) = ∏_{p|M}(1 − p^{−1})L_{M}(2s, f, ∨^{2})ζ_{M}(6s − 1).

Here L_{M}(2s, f, ∨^{2}), ζ_{M}(6s − 1) denote the symmetric square L-function of f at 2s and the zeta function at 6s − 1, each with Euler factors corresponding to primes dividing M removed.

Proof: Taking real parts of w, s sufficiently large to guarantee absolute convergence we apply the basic identity translated into the equality of equations (3.7) and (3.8). It follows that for the real part of s sufficiently large

lim_{w→1}(w − 1)Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})

= ∑_{(n,M)=1,n>0} \frac{lim_{w→1}(w − 1)L_{M}(w, χ̃_{n_{0}}χ_{a_{2}l_{2}})χ_{a_{1}l_{1}}(n_{0})c(n_{0}n_{1}^{2})Q_{n_{0},n_{1}}^{(a_{2}l_{2})}(w)}{n^{s}}.

Now lim_{w→1}(w−1)L_{M}(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) vanishes unless χ_{n_{0}}χ_{a_{2}l_{2}} is the trivial character. This can only happen if n_{0} = 1 and a_{2}l_{2} = 1. Thus the residue vanishes unless a_{2}l_{2} = 1. If a_{2}l_{2} = 1 then all the terms on the right hand side of the above will vanish unless n_{0} = 1, in which case L_{M}(w, χ̃_{n_{0}}χ_{a_{2}l_{2}}) = ζ_{M}(w) and the residue is ∏_{p|M}(1 − p^{−1}).

After taking the residue, the right hand side of the above reduces to

∏_{p|M}(1 − p^{−1}) ∑_{(n_{1},M)=1,n_{1}>0} \frac{c(n_{1}^{2})Q_{1,n_{1}}^{(1)}(1)}{n_{1}^{2s}}.

Referring to (2.7) we see that each p-part sums to the p-part of

L_{M}(2s, f, ∨^{2}) ζ_{M}(6s − 1)

as claimed.

Recall now that the product

A(s, w)A(α(s, w))A(β(s, w))A(βα(s, w))B(s, w)B(α(s, w))P(s, w)× Z_{M}(s, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}}) was shown in Proposition 3.5 to be an entire function of s, w. Specializing to w = 1 and removing the w − 1 factor, the remaining factors of P(s, 1) are (3s − 3/2)(3s − 1)(3s − 1/2). Thus the only possible poles of the residue R_{M}(s; χ_{a_{1}l_{1}}) are at zeros of A(s, 1)A(α(s, 1))A(β(s, 1))A(βα(s, 1))B(s, 1)B(α(s, 1)) or at s = 1/2, 1/3, 1/6. The possible poles corresponding to zeros of Euler factors of A, B are eliminated as in Proposition 3.6. The factor 3s − 1 cancels the simple pole of ζ_{M}(6s − 1) at s = 1/3. The possible pole at s = 1/6 is easily eliminated and we are left with the possibility of a simple pole at s = 1/2. This will occur if and only if L_{M}(2s, f, ∨^{2}) has a simple pole at s = 1/2. We have thus completed the proof of the first part of Theorem 0.3.

We now proceed with the proof of a mean value result from which the second half of Theorem 0.3 will follow. We will work with the case that L_{M}(2s, f, ∨^{2}) has a simple pole at s = 1/2 and denote the non-zero residue as R_{M}, so

(3.22)  R_{M} = lim_{s→1/2}(s − 1/2)L_{M}(2s, f, ∨^{2}).

The polar line w = 1 is reflected into the lines w = −3s + 5/2 and 2w = −3s + 3 by the functional equations α, β. The third of these gives a single polar line passing through (1/2, 3/4) while w = 1, w = −3s + 5/2 give two polar lines intersecting at (1/2, 1). The behavior of Z_{M}(s, w; 1, χ_{a_{1}l_{1}}) near (1/2, 1) is given by a sum of contributions from the two polar lines:

(3.23) Z_{M}(s, w; 1, χ_{a_{1}l_{1}}) =

\frac{A_{0}}{(w − 1)(s − 1/2)} + \frac{A_{1}(s)}{w − 1} + \frac{A′_{0}}{(w + 3s − 5/2)(s − 1/2)} + \frac{A′_{1}(s)}{w + 3s − 5/2} + H(s, w),

where

A_{0} = lim_{s→1/2} lim_{w→1}(s − 1/2)(w − 1)Z_{M}(s, w; 1, χ_{a_{1}l_{1}}),

A′_{0} = lim_{s→1/2} lim_{w→5/2−3s}(s − 1/2)(w + 3s − 5/2)Z_{M}(s, w; 1, χ_{a_{1}l_{1}}),

and H(s, w) is an analytic function of s, w near (1/2, 1). Let s → 1/2 in (3.23) while w ≠ 1. We know from the left hand side of (3.23) that this must remain analytic. However, summing the two leading fractions, we see that this can only happen if A′_{0} = −A_{0}, in which case (3.23) reduces, at s = 1/2, to

(3.24)  Z_{M}(1/2, w; 1, χ_{a_{1}l_{1}}) = \frac{3A_{0}}{(w − 1)^{2}} + \frac{B_{1}}{w − 1} + I(w),

where B_{1} is computable and I(w) is an analytic function of w near w = 1.

We will now state the mean value theorem and do the remaining bit of work necessary to complete the proof.

Theorem 3.8. Let l_{1}, l_{2} > 0, l_{1}, l_{2}|M and a_{1}, a_{2} ∈ {1, −1} and let the L-series L_{M}(s, f, χ_{d_{0}}χ_{a_{1}l_{1}}) and correction factors P_{a_{1}l_{1}d_{0},d_{1}}(s) be as defined above. Then

∑_{d>0,(d,M)=1} L_{M}(1/2, f, χ_{d_{0}}χ_{a_{1}l_{1}})χ_{a_{2}l_{2}}(d_{0})P_{a_{1}l_{1}d_{0},d_{1}}(1/2)e^{−d/x}

= δ_{1,a_{2}l_{2}} ∏_{p|M}(1−p^{−1})R_{M}ζ_{M}(2)(x log x)+δ_{1,a_{2}l_{2}}C_{M}x+C′_{M}x^{3/4}+C′′_{M} +O(x^{−3/4}).

Here R_{M} = lim_{s→1/2}(s−1/2)L_{M}(2s, f, ∨^{2}), C_{M}, C′_{M}, C′′_{M} are computable constants and δ_{1,a_{2}l_{2}} = 1 if a_{2}l_{2} = 1 and is 0 otherwise.

Proof: Applying the integral transform

\frac{1}{2πi} ∫_{2−i∞}^{2+i∞} Γ(w)x^{w}dw = e^{−1/x},

valid for x > 0, we obtain first

\frac{1}{2πi} ∫_{2−i∞}^{2+i∞} Z_{M}(1/2, w; χ_{a_{2}l_{2}}, χ_{a_{1}l_{1}})Γ(w)x^{w}dw =

∑_{d>0,(d,M)=1} L_{M}(1/2, f, χ_{d_{0}}χ_{a_{1}l_{1}})χ_{a_{2}l_{2}}(d_{0})P_{a_{1}l_{1}d_{0},d_{1}}(1/2)e^{−d/x}.

Moving the line of integration to ℜ(w) = −3/4 − ϵ, for ϵ > 0, we pick up from the pole at w = 1 a polynomial of the form x(3A_{0} log x + C_{M}), where the constants A_{0}, C_{M} are computable and

(3.25)  A_{0} = lim_{s→1/2}(s − 1/2)R_{M}(s; χ_{a_{1}l_{1}}) = ∏_{p|M}(1 − p^{−1})R_{M}ζ_{M}(2).

Passing the pole at w = \frac{3}{4} we pick up an additional contribution of C′_{M}x^{3/4} and at w = 0 we get a contribution of C′′_{M}. The integral at ℜ(w) = −3/4 − ϵ converges absolutely by the upper bound estimate of Proposition 3.6 (as Γ(w) decays exponentially), and contributes an error on the order of x^{−3/4−ϵ}. This completes the proof of Theorem 3.8.

The remainder of Theorem 0.3 follows after taking a linear combination of different χ_{a_{1}l_{1}}, χ_{a_{2}l_{2}} to sieve out d_{0} with particular quadratic residue restrictions. As the leading term in the mean value estimate is independent of a_{1}l_{1} and vanishes if a_{2}l_{2} ≠ 1, for any class of d satisfying the restrictions of Theorem 0.3 there will be a non-zero leading term in a sum over this restricted class.

It follows from the explicit description (2.1–2.3) that P_{a_{1}l_{1}d_{0},d_{1}}(1/2) ≪_{d_{0},ϵ} d_{1}^{2η+ϵ} where, as mentioned previously, c(m) ≪_{ϵ} m^{η+ϵ} is any valid upper bound for the coefficients of the GL(3) form f. As a consequence, the correction factors P_{a_{1}l_{1}d_{0},d_{1}}(1/2) attached to a single d_{0} can only add a contribution on the order of x^{1/2+η+ϵ}. As the bound η = 2/5 is true, and 1/2 + 2/5 < 1, there must be non-vanishing for infinitely many distinct d_{0}.

Let us remark that since the symmetric square L-function has a pole, the sum of the quadratic twists is on the order of x log x, as we have seen in Theorem 3.8, with no conditions at all on the signs of the functional equations. By contrast, if there were no pole, then the corresponding sums would be on the order of x, and could conceivably lack a main term if two colliding main terms had opposite signs. Thus to obtain a nonvanishing theorem the precise analysis in that case is more delicate, and we do not carry it out here.

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