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On L-functions associated with the vector space of binary quadratic forms

Published online by Cambridge University Press:  22 January 2016

Hiroshi Saito
Hiroshi Saito*
Affiliation:
Graduate School of Human and Environmental Studies, Kyoto UniversityKyoto 606, Japan
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The purpose of this paper is to prove functional equations of L-functions associated with the vector space of binary quadratic forms and determine their poles and residues. For a commutative ring K, let V(K) be the set of all symmetric matrices of degree 2 with coefficients in K. In V(C), we consider the inner product

where for .

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 130 , June 1993 , pp. 149 - 176
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Arakawa, T., Special values of L-functions associated with the space of quadratic forms and the representation of Sp(2n, F p ) in the space of Siegel cusp forms, Automorphic forms and geometry of arithmetic varieties, Advanced Studies in Pure Math, 15, 465–508, Kinokuniya, 1989.Google Scholar
[2] Bernstein, I.N. and Gelfand, S.I., Meromorphic property of the function P , Funct. Anal. Appl, 3 (1969), 68–69.Google Scholar
[3] Gelfand, I.M. and Shilov, G.E., Generalized functions, vol. 1, Academic Press, New York, 1964.Google Scholar
[4] Saito, H., A generalization of Gauss sums and its applications to Siegel modular forms and L-functions associated with the vector space of quadratic forms J. reine angew. Math., 416 (1991), 91–142.Google Scholar
[5] Sato, F., On zeta functions of ternary zero forms, J. Fac. Sci. Univ. Tokyo, 28 (1982), 585–604.Google Scholar
[6] Sato, F., On functional equation of zeta distribution, Automorphic forms and geometry of arithmeic varieties, Advanced Studies in Pure Math., 15, 465–508, Kinokuniya, 1989.Google Scholar
[7] Sato, M., Theory of prehomogeneous vector spaces (notes by T. Shintani in Japanese), Sugaku no Ayumi, 15 (1970), 85–157.Google Scholar
[8] Shintani, T., On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms, J. Math. Soc. Japan, 24 (1972), 132–188.CrossRefGoogle Scholar
[9] Shintani, T., On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, 22 (1975), 25–65.Google Scholar