The modularity of Siegel’s zeta functions

Siegel deﬁned zeta functions associated with indeﬁnite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coe ﬃ cients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the e ﬀ ect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel’s original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that “half” of Siegel’s zeta functions correspond to holomorphic modular forms.


Introduction
In 1903, Epstein [3] defined the zeta function ζ 0 (s) associated with a positive definite symmetric matrix S of degree m by and studied their analytic properties such as analytic continuations and functional equations.(For a modern treatment of Epstein's zeta functions, we refer to Terras [31, §1.4.2].)In a 1938 paper [23], Siegel defined and investigated the zeta functions associated with quadratic forms of signature (1, m − 1), and in a 1939 paper [24], those for general indefinite quadratic forms.Although Siegel's calculations were rather involved, Siegel's results are now well understood in the framework of the theory of prehomogeneous vector spaces.Let Y be a non-degenerate half-integral symmetric matrix of degree m with p positive eigenvalues and m − p negative eigenvalues (0 < p < m).Let S O(Y) be the special orthogonal group of Y and denote by S O(Y) Z its arithmetic subgroup.We put V ± = {v ∈ R m ; sgn Y[v] = ±}.Then Siegel's zeta functions are Dirichlet series associated with the prehomogeneous vector space (GL 1 (C) × S O(Y), C m ), and are defined by where the sum runs over a complete set of representatives of S O(Y) Z \(Z m ∩ V ± ), and µ(v) is a certain volume of the fundamental domain related to the isotropy subgroup S O(Y) v of S O(Y) at v. In the positive definite case, the modularity of Epstein's zeta function ζ 0 (s) is almost obvious; ζ 0 (s) is obtained by taking the Mellin transform of (the restriction to the imaginary axis of) the theta series which is a modular form for a subgroup of S L 2 (Z).(cf.Miyake [15, §4.9], Terras [31, §3.4.4].)On the contrary, it is not clear from the definition whether or not Siegel's zeta function arises as an integral transform of some infinite series with modular properties.Rather, in the preface to a 1938 paper [23], Siegel wrote that such theta series would be constucted from his zeta functions, citing the work of Hecke [7], in which Hecke derived the transformation formula for the theta series associated with indefinite binary quadratic forms from the functional equation of zeta functions of real quadratic fields.Furthermore, Siegel made the following remark in the last section of [23]: Will man die Transformationstheorie von f (S, x) für beliebige Modulsubstitutionen entwickeln, so hat man außer ζ 1 (S, s) auch analog gebildete Zetafunktionen mit Restklassen-Chrakteren zu untersuchen.Die zum Beweise der Sätze 1, 2,3 führenden Überlegungen lassen sich ohne wesentiche Schwierigkeit auf den allgemeinen Fall übertragen.Vermöge der Mellinschen Transformation erhält man dann das wichtige Resultat, daß die durch (53) definierte Funktion f (S, x) eine Modulform der Dimension n 2 und der Stufe 2D ist; dabei wird vorausgesetzt, daß n ungerade und x ′ Sx keine ternäre Nullform ist.
If one wants to develop the transformation theory of f (S, x) for arbitrary modular substitutions, then in addition to ζ 1 (S, s) one also has to investigate zeta functions formed analogously with residual class characters.The considerations leading to the proof of Theorems 1, 2, 3 can be transferred to the general case without any major difficulty.By virtue of the (inverse) Mellin transformation, one then obtains an important result that the function f (S, x) defined by ( 53) is a modular form of weight n 2 and level 2D, provided that n is odd and x ′ Sx is not a ternary zero form.
As of 1938, Siegel seemed to have noticed the possibility that by considering the twists of zeta functions by Dirichlet characters, one can prove modularity for congruence subgroups.In the holomorphic case, this fact is known as Weil's converse theorem [34].It was 1967 when Weil's paper [34] appeared!Revisiting Siegel's prediction in the light of recent developments is one motivation for the present study.
We should note that in the quotation above, Siegel mentioned the parity of n, the number of variables of quadratic forms.This is related to the fact that the concept of nonholomorphic modular forms was not yet in place at that time.In a celebrated paper [12], Maaß introduced the notion of the so-called Maass forms and established a Hecke correspondence for Maass forms.Further, in [13], as its application of his theorem, Maaß proved that in a very special case (when Y is diagonal of even degree with det Y = 1), Siegel's zeta functions can be expressed as the product of two standard Dirichlet series such as the Riemann zeta function ζ(s) and the Dirichlet L-function L(s, χ).On the other hand, it is only recently that papers on Weil-type converse theorems for Maass forms have emerged (cf.[16,17]).It would be a very natural idea for us to accomplish Siegel's original plan to prove the modularity via converse theorem including the case of nonholomorphic forms.
Siegel's zeta functions are closely related to the so-called Siegel's main theorem (Siegelsche HauptSatz).In a 1951 paper [26], Siegel proved the transformation formula for some theta series arising from indefinite quadratic forms, and the equality between an integral of the indefinite theta series over fundamental domains and some Eisenstein series (cf.[26,Satz 1]).It was shown in [26,Hilfssatz 4] that the coefficients of ζ ± (s) coincide with the Fourier coefficients of the non-holomorphic modular forms appearing in Siegel's formula.Here we ignore the differences in the definitions of µ(v); the definitions of measures are different for each of the papers [23,24,26].Siegel called M(Y; n) the measures of representations (Darstellungsmaß).The measure M(Y; n) of representations is an analogue of the representation number for a positive symmetric matrix S , and Siegel's formula can be reformulated as an arithmetic identity that M(Y, n) is equal to the product of local representation densities over all primes.Weil [33] generalized Siegel's result by using the language of adeles, and it is the Siegel-Weil formula-a cornerstone in the modern number theory.Now we explain the main results of the present paper.First, along the Sato-Shintani theory [21] of prehomogeneous vector spaces, we define Siegel's zeta functions and prove their analytic properties.Here, to treat twisted zeta functions as well as the original Siegel's zeta functions, we first consider Siegel's zeta functions with congruence conditions, which are defined using Schwartz-Bruhat functions on Q m .This idea is due to F. Sato [20].Then the converse theorem in [16] is applied to the zeta functions, and the following result is obtained: Main result 1 (Theorem 2).Let m ≥ 5. Assume that at least one of m or p is an odd integer.Take an integer ℓ with ℓ ≡ 2p − m (mod 4), and put D = det(2Y).Let N be the level of 2Y.Define C ∞ -function F(z) = F(x + iy) on the Poincaré upper half-plane H by where d 1 g is a suitably normalized Haar measure on S O(Y) R , α(0) is some constant determined by the residues of ζ ± (s), and W µ,ν (y) denotes the Whittaker function.Then, F(z) is a Maass form of weight ℓ/2 with respect to the congruence subgroup Γ 0 (N).
The above formula can be compared with Siegel's calculation [26,Hilfssatz 4] of the Fourier expansions of non-holomorphic modular forms.Our F(z) is essentially the same as the modular form given by Siegel [26].See Remark 4. The theorem above excludes the case where both m and p are even.Our second result states that if one of m − p and p are even, we can construct holomorphic modular forms from M(Y; ±n).
Main result 2 (Theorem 3).Let m ≥ 5. Assume that m − p is even.We define a holomor-phic function F(z) on H by Then, F(z) is a holomorphic modular form of weight m/2 with respect to Γ 0 (N).(In the case that p is even, we can construct holomorphic modular forms from M(Y; The theorem above is consistent with a result of Siegel that was published in a 1948 paper [25].In this paper, Siegel calculated the action of certain differential operators on indefinite theta series, and proved that in the case of det Y > 0, we can construct holomorphic modular forms from indefinite theta series associated with Y. Before closing Introduction, we give some remarks on related researches, future problems, and possible applications.Special values of Siegel's zeta functions associated with Y of signature (1, m − 1) appear in the dimension formula for automorphic forms on orthogonal groups of signature (2, m) (cf.Ibukiyama [8]), and it is important to investigate their arithmetic aspects.Ibukiyama [9] proved an explicit formula expressing Siegel zeta functions (with m even) as linear combinations of products of two shifted Dirichlet Lfunctions and certain elementary factors.His proof is given by direct calculations using Siegel's main theorem in [26] and not by converse theorems.Ibukiyama's explicit formula is quite general and includes the above-mentioned result of Maaß [13].We also mention the work [6] of Hafner-Walling, in which they carried out extensive calculations to make Siegel's formula more explicit in terms of standard Eisenstein series.This work is also restricted to the case where m is even.It is worthwhile to investigate the case where m is odd.Finally, in a good situation, the method of converse theorems can be used to prove lifting theorems.In [29], a Shintani-Katok-Sarnak type correspondence is derived from analytic properties of a certain prehomogeneous zeta function whose coefficients involve periods of Maass cusp forms.In [14], Maaß studied a generalization of Siegel's zeta functions, which can be regarded as prehomogeneous zeta functions whose coefficients involve periods of automorphic forms on orthogonal groups.It is quite probable that our method can be applied to these zeta functions, and some lifting theorems will be obtained.We hope to discuss this topic elsewhere.
The present paper is organized as follows.In Section 1, we recall a Weil-type converse theorem for Maass forms, and in Section 2, we define our prehomogeneous vector spaces and give the local functional equations.Section 3 is devoted to define Siegel's zeta functions with congruence conditions, and analytic properties of Siegel's zeta functions are proved in Section 4. We prove our main theorems in Sections 5 and 6.
Acknowledgement.The author wishes to thank Professor Fumihiro Sato for stimulating discussion and helpful suggestions.The author also thanks Professor Tomoyoshi Ibukiyama for valuable comments; in particular, Professor Ibukiyama explained the relation of the results of this paper to the prior work of [8,9,13,25].Finally, the author would like to thank anonymous reviewers for their careful reading and helpful comments.
Notation.We denote by Z, Q, R, and C the ring of integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively.The set of non-zero real numbers and the set of positive real numbers are denoted by R × and R + , respectively.The set of positive integers and the set of non-negative integers are denoted by Z >0 and Z ≥0 , respectively.The real part and the imaginary part of a complex number s are denoted by ℜ(s) and ℑ(s), respectively.For complex numbers α, z with α 0, α z always stands for the principal value, namely, α z = exp((log |α| + i arg α)z) with −π < arg α ≤ π.We use e[x] to denote exp(2πix).The quadratic residue symbol * * has the same meaning as in Shimura [22, p. 442].For a meromorphic function f (s) with a pole at s = α, we denote its residue at s = α by Res s=α f (s).

A Weil-type converse theorem for Maass forms
In this section, we define Maass forms on the Poincaré upper half-plane H = {z ∈ C | ℑ(z) > 0} of integral and half-integral weight, and recall a Weil-type converse theorem for Maass forms that is proved in [16].We refer to Cohen-Strömberg [2] for an overview of the theory of Maass forms.Let Γ = S L 2 (Z) be the modular group, and for a positive integer N, we denote by Γ 0 (N) the congruence subgroup defined by As usual, Γ acts on H by the linear fractional transformation We put j(γ, z) = cz + d, and define θ(z) and J(γ, z) by Then it is well-known that where For an integer ℓ, the hyperbolic Laplacian ∆ ℓ/2 of weight ℓ/2 on H is defined by Let χ be a Dirichlet character mod N. Then we use the same symbol χ to denote the character of Γ 0 (N) defined by ( 3) Definition 1 (Maass forms).Let ℓ ∈ Z, and N be a positive integer, with 4|N when ℓ is odd.A complex-valued C ∞ -function F(z) on H is called a Maass form for Γ 0 (N) of weight ℓ/2 with character χ, if the following three conditions are satisfied; (i) for every γ ∈ Γ 0 (N), (iii) F is of moderate growth at every cusp, namely, for every A ∈ S L 2 (Z), there exist positive constants C, K and ν depending on F and A such that We call Λ the eigenvalue of F.
We put and denote by ψ r,0 the principal character modulo r.Recall that the Gauss sums are calculated as follows: Let P N be a set of odd prime numbers not dividing N such that, for any positive integers a, b coprime to each other, P N contains a prime number r of the form r = am + b for some m ∈ Z >0 .For an r ∈ P N , denote by X r the set of all Dirichlet characters mod r (including the principal character ψ r,0 ).For ψ ∈ X r , we define the Dirichlet character ψ * by (12) ψ We put (For the definition of ε r , see (1).) In the following, we fix a Dirichlet character χ mod N that satisfies χ(−1) = i ℓ (resp.χ(−1) = 1) when ℓ is even (resp.odd).

Prehomogeneous vector spaces
Let Y be a non-degenerate half-integral symmetric matrix of degree m, and let p be the number of positive eigenvalues of Y. Throughout the present paper, we assume that m ≥ 5 and p(m − p) > 0. We denote by S O(Y) the special orthogonal group of Y defined by Let P(v) be the quadratic form on V defined by ( 17) where we use Siegel's notation.Then, for g = (t, g) ∈ G and v ∈ V, we have ( 18) and V − S is a single ρ(G)-orbit, where S is the zero set of P: That is, (G, ρ, V) is a reductive regular prehomogeneous vector space.(We refer to [11,21] for the basics of the theory of prehomogeneous vector spaces.)We identify the dual space V * of V with V itself via the inner product v, v * = t vv * .Then the dual triplet (G, ρ * , V * ) is given by We define the quadratic form Then, for g and V − S * is a single ρ * (G)-oribit, where S * is the zero set of P * : For ǫ, η = ±, we put We denote by dv = dv 1 • • • dv m the Lebesgue measure on V R , and by S(V R ) the space of rapidly decreasing functions on V R .Then, for f, f * ∈ S(V R ) and ǫ, η = ±, we define the local zeta functions Φ ǫ ( f ; s) and Φ * η ( f * ; s) by ( 21) For ℜ(s) > m 2 , the integrals Φ ǫ ( f ; s) and Φ * η ( f * ; s) converge absolutely, and as functions of s, they can be continued analytically to the whole s-plane as meromorphic functions.Further, we define the Fourier transform The following lemma is due to Gelfand-Shilov [5]; a detailed proof is given in Kimura [11, § 4.2].

Lemma 2 (Local Functional Equation). Let p be the number of positive eigenvalues of Y, and put D = det(2Y). Then the following functional equation holds:
In the rest of this section, we investigate singular distributions whose supports are contained in the real points S R of S ; these distributions play an important role in the calculation of residues of Siegel's zeta functions.We decompose S R as For i = 1, . . ., m, we define an (m − 1)-dimensional differential form ω i on U i by ( 22) It is easy to see that there exists an (m − 1)-dimensional differential form ω on S R that satisfies Since P(gv for g ∈ S O(Y) R and t > 0. Similary, for the zero set S * of P * , we decompose the real points S * R as The same argument as above ensures the existence of an (m − 1)-dimensional differential form ω * on S * 1,R such that the restriction of ω * on is given by We have This is stated, without proof, on p. 156 of Sato-Shintani [21] where Siegel's zeta function is picked up as an example of their theory.Since the details cannot be found in other literature, we give a proof of the lemma for convenience of readers.
We may replace From the identity (19) (or the first formula on p. 257) in Gelfand-Shilov [5, Chap III, §2.2], we have By the shift s → s − m 2 , we have It then follows from the local functional equation (Lemma 2) that lim , which proves the first assertion of Lemma 3. The second assertion can be proved in a similar fashion.

Siegel's zeta functions with congruence conditions
In this section, followig M. Sato-Shintani [21], we define Siegel's zeta functions associated with (G, ρ, V), and give their integral representations.Moreover, we calculate the singular parts of the zeta integrals.For this calculation, we also refer to Kimura [11].Furthermore, following F. Sato [20], we slightly generalize Siegel's zeta functions with using Schwartz-Bruhat functions on Q m in order to treat the twisted zeta functions simultaneously.Let dx be the measure on GL m (R) defined by and dλ the measure on the space Sym m (R) of symmetric matrices of degree m defined by Then we normalize a Haar measure d 1 g on the Lie group S O(Y) R in such a way that the integration formula ( 25) holds for all integrable functions F(x) ∈ L 1 (GL m (R)).Further, let dt be the Lebesgue measure on R and put (26) holds for all integrable functions H(t, g) ∈ L 1 (G R ).Similarly, for v * ∈ V * η , we write and fix a Haar measure dµ * v * on S O(Y) v * ,R such that the integration formula (28) holds for all integrable functions H(t, g) ∈ L 1 (G R ).
We call a function φ : V Q → C a Schwartz-Bruhat function if the following two conditions are satisfied: (1) there exists a positive integer M such that φ(v) = 0 for v 1 M V Z , and (2) there exists a positive integer The totality of Schwartz-functions on V Q is denoted by S(V Q ).We define the Fourier transform φ ∈ S(V Q ) of a Schwartz-Bruhat function φ ∈ S(V Q ) by ( 29) where r is a sufficiently large positive integer such that the value φ(v)e[− v, v * ] depends only on the residue class v mod rV Z .Though r is not unique, the value φ(v * ) does not depend on the choice of r.The following lemma is essentially an adelic version of Poisson summation formula.
we have the following In the following, we assume that Then we define the zeta integral Z( f, φ; s) by ( 31) we have, by a formal calculation, and further, by applying (27) to we have In the following, for v Since it is assumed that m ≥ 5, the generic isotropy subgroup S O(Y) v is a semisimple algebraic group, and thus we have µ(v) < +∞.(cf.[11, p. 184].)We further put ρ(t, ġ)v = x in the right hand side above.Then, since The Dirichlet series converges absolutely for ℜ(s) > m 2 , as will be explained in Remark 2 shortly.Hence the interchange of summation and integration, which leads to (33), can be justified under this condition.Similary, for f * ∈ S(V R ) and φ * ∈ S(V Q ) that satisfies (34) φ we define the zeta ingegral Z * ( f * , φ * ; s) by where dµ * v * is the Haar measure on S O(Y) v * ,R defined by (28).Definition 2 (Siegel's zeta functions with congruence conditions).Let ǫ, η = ± and assume that φ, φ * ∈ S(V Q ) satisfy (30), (34), respectively.Then we define ζ ǫ (φ; s) and ζ * η (φ * ; s) by We can summarize our argument as the following Lemma 6 (Integral representations of the zeta functions).Let f, f * ∈ S(V R ) and assume that φ, φ The original Siegel's zeta functions are obtained by letting φ = φ 0 , where φ 0 is the characteristic function ch V Z of V Z .To apply Weil-type converse theorems, we need to examine the case where φ(v) = ψ(P(v))φ 0 (v) with Dirichlet character ψ.Since each φ(v) is a linear combination of characteristic functions of subsets of the form a + NV Z (a ∈ V Q , N ∈ Z ≥1 ), we call ζ ǫ (φ; s), ζ * η (φ * ; s) Siegel's zeta functions with congruence conditions.
(2) The absolute convergence of Siegel's zeta functions is not at all obvious, though Siegel wrote just "Die Konvergents der Reihe entnimmt man der Reduktiontheorie".A detailed proof of the convergence can be found in Tamagawa [30].It also follows from the general theory of prehomogeneous vector spaces (Saito [18], F. Sato [19]).
(3) We can write ζ ± (φ; s) as Since φ(v) = 0 for v 1 L V Z with some integer L, we see that that the sum in the definition of M(P, φ; ±r) is a finite sum (cf.Kimura [11,p.184]).In the case of φ = φ 0 , we have supp(φ 0 ) = V Z and P(v) ∈ Z \ {0} for v ∈ V ± ∩ V Z .For n = 1, 2, . . ., we put (39) Siegel called M(P; n) the measures of representation (Darstellungsmaß).We have To investigate analytic properties of the zeta integrals, we define measures on isotropy subgroups at singular points.We fix an arbitrary point v of S 1,R .Recall that in the previous section, we have defined an We can normalize a measure dσ v on the isotropy subgroup S O(Y) v,R in such a way that the integration formula (40 holds for all integrable functions ψ(g) ∈ L 1 (S O(Y) R ).Similarly, for v * ∈ S * 1,R , we take a measure dσ * v * on the isotropy subgroup S O(Y) v * ,R such that the integration formula (41 holds for all integrable functions ψ(g) ∈ L 1 (S O(Y) R ).Now we put It is obvious that The four integrals above converges absolutely for ℜ(s) > m 2 , and further, two integrals Z + ( f, φ; s) and Z * + ( f * , φ * ; s) are absolutely convergent for any s ∈ C and define entire functions of s.Let us calculate Z − ( f, φ; s) formally by using Lemma 5, the Poisson summation formula; the interchange of integral and summation will be justified later in Remark 3. Since χ(t, g) = χ * (t, g) −1 = t 2 , it follows from Lemma 5 that The first term of the most right hand side is 1 0 Using (40) and (41), we calculate the second and third terms following the method of Sato-Shintani [21, Theorem 2].Put By the interchange of summation and integration, the third term above becomes Here we have used (23) in the third equality.Hence the integral (42) is calculated as Similarly, by term-by-term integration, we have and by using ( 24) and (41), we obtain Hence we see that Then we have the first assertion of the following lemma; the second assertion can be proved similarly as the first assertion, and then the third assertion follows immediately from the first and second assertions.
(3) As functions of s, the integrals Z( f, φ, s) and Z * ( f , φ; s) can be continued analytically to the whole s-plane, and satisfy the following functional equation: Remark 3. In [10], Igusa studied the so-called admissible representations related to the Siegel-Weil formula [33].According to his classification, our prehomogeneous vector space (GL 1 (C) × S O(Y), C m ) gives an admissible representation if m ≥ 5, and this implies that the integrals are absolutely convergent for all Schwartz-Bruhat functions f, f * ∈ S(V R ) and φ, φ * ∈ S(V Q ).Hence the integrals which appear in Lemma 7, are absolutely convergent, and the interchange of integral and summation can be justified by Fubini's theorem.
4 Analytic properties of Siegel's zeta functions (1) The zeta functions ζ ǫ (φ; s) and ζ * η ( φ; s) have analytic continuations of s in C, and the zeta functions multiplied by (s − 1)(s − m 2 ) are entire functions of s of finite order in any vertical strip.
(2) Th zeta functions ζ ǫ (φ; s) and ζ * η ( φ; s) satisfy the following functional equation: (3) The residues of ζ ǫ (φ; s), ζ * η ( φ; s) at s = 1 and s = m 2 are given by The following relations hold: . Then we see that and thus the integral Z( f, φ; s) can be continued to a meromorphic function on the whole C, and (s − 1)(s − m 2 )Z( f, φ; s) is an entire function of s.Further, for any s ∈ C, we take and by Lemma 2, we have where A(s) is given by This implies that the vector (53) For any s ∈ C, there exists an f ǫ ∈ C ∞ 0 (V ǫ ) such that Φ ǫ ( f ; m 2 −s) 0, and hence (53) is the zero vector.This proves the functional equation ( 46).Next we calculate the residues.For the simple pole at s = m 2 , we have Res

and similarly
Res By Lemma 7 (1), it is easy to pick up the residue of Z( f, φ; s) at the simple pole s = 1, and together with Lemma 6, it implies that for Here the value Φ ǫ ( f ; 1) is meaningful, and Furthermore, by Lemma 3 (1), we have , and hence we obtain the residue formula (49).Similarly, the residue formula (50) can be proved with Lemma 3 (2); the detail is omitted.To prove the relation (51), we let s = 1 in the functiona equation (46): By using (50) and we see that we obtain the desired relation Finally, let s = m 2 − 1 in the functional equation (46).We have and by using (49), we obtain Let N be the level of 2Y.By definition, N is the smallest positive integer such that N(2Y) −1 is an even matrix (a matrix whose entries are integers and even along the diagonal).We normalize the zeta functions ζ ǫ (φ; s), ζ * η ( φ; s) as follows: Lemma 8.The normalized zeta functions ζ ǫ (φ; s), ζ * η ( φ; s) satisfy the following functional equation: where γ(s) and Σ(ℓ) are matrices defined by (4).
The functional equation ( 56) is quite the same as the functional equation of the condition [A3] in § 1 with m 2 = 2λ, ℓ ≡ 2p − m (mod 4).Hence it is reasonable to expect that our converse theorem (Lemma 1) can apply to the normalized zeta functions ζ ǫ (φ; s), ζ * η ( φ; s) to obtain Maass forms.The following lemma is indispensable for the application.(2) Assume that m is even and p is odd.Let q = m 2 .Then we have Proof.By a little calculation, we obtain (58) Let us consider the values of both sides at s = −k (k ∈ Z >0 ).On the left hand side, ), and if m is even, then Γ(s)Γ s + m 2 − 1 = Γ(s)Γ(s + q − 1) has a simple pole at s = −k (1 ≤ k ≤ q − 2).We assume that 1 ≤ k ≤ q − 2 in the case of even m.Then, since Γ(s)Γ s + m 2 − 1 has a simple pole at s = −k, we see that If m is odd, then either p or m − p is odd, and thus we have In the case of even m, if p is odd, then the relation above should hold.In the case that both of p and m − p are even, this argument can not apply since sin πp 2 = sin π(m−p) 2 = 0.
The following lemma follows immediately from the relations (51) and (52).
Lemma 10.We have the following relations: In the rest of this section, we discuss the invariance of volumes with respect to scalar multiplications.
(2) Let us show that σ(rv) = r 2−m • σ(v).Take an f ∈ S(V R ) and put ψ(g) = f (gv).By using (40), we have By the substitution v → rv, we have where we have used (23) on the fifth equality.This proves σ(rv) = r 2−m • σ(v).The second formula can be proved in a similar fashion.

The main theorem
To prove the functional equation of twisted zeta functions, we quote a result of Stark [28].
Let Y be a non-degenerate half-integral symmetric matrix of degree m.Let D = det(2Y) and N be the level of 2Y.We define a half-integral symmetric matrix Y by We define the quadratic form P(v) on V by P(v) = Y[v] = t vYv, and the quadratic form where P * is defined by (19).For this P, we define the measure M * ( P; n) of representation by For an odd prime r with (r, N) = 1 and a Dirichlet character ψ of modulus r, we define the function φ ψ,P (v) on V Q by φ ψ,P (v) = τ ψ (P(v)) • φ 0 (v), where τ ψ (P(v)) is the Gauss sum defined by (9), and φ 0 (v) is the characteristic function of Z m .It is easy to see that φ ψ,P (v) is a Schwartz-Bruhat function on V Q .We define a field K by , and χ K be the Kronecker symbol associated to K. (If K = Q, we regard χ K as the principal character.)Furthermore, we define a Dirichlet character ψ * mod r by and put as (13).Then the following lemma follows from Stark [28, Lemmas 5 and 6].
Then, F(z) (resp.G(z)) is a holomorphic modular form for Γ 0 (N) of weight m/2 with character χ K (resp.χ K N ).Further we have Remark 5.If p is even, we can prove the same assertion for M(P; −n).Theorem 2 excludes the case where both m and p are even, but Theorem 3 shows that both ζ + and ζ − correspond to holomorphic modular forms in this case.
[188][189], the number of S O(Y) Z -orbits in primitive vectors in S 1,Z and S * 1,Z is finite.Lemma 14. (1) We call a vector v = (v 1 , . . ., v m ) ∈ V Z primitive if the greatest common divisor of v 1 , . . ., v m is 1.Then {v ∈ S O(Y) Z \S 1,Z ; vis primitive} is a finite set.Let a 1 , . . ., a h be a complete system of representatives of this set.Then we have v∈S O(Y) Z \S 1,Z σ(v) = ζ(m − 2) h i=1 σ(a i ).