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Abstract. In this article we study a Rankin-Selberg convolution of two complex variables attached to Siegel modular forms of degree 2. We establish its basic analytic properties, find its singular curves and compute some of its residues. In particular, we show that two known Dirichlet series of Rankin-Selberg type, one introduced by Maass and another by Kohnen and Skoruppa, are obtained as residues from this series of two variables. Furthermore, we define and study a collection of Rankin-Selberg convolutions for Jacobi forms of degree 1. 1991 Mathematics Subject Classification: 11F46, 11F55, 11M41. 1 Introduction À nÁ Let wn ðmÞ ¼ m be the quadratic character of conductor n, p an automorphic representation and Lðw; pÞ the corresponding L-function. The Dirichlet series in two complex variables Dðs; wÞ ¼ P Lðw; p; wn Þ ns n>0 has been the object of some considerable interest. (See for example [3], [4], [5], [8], [9], [16]) The analytic properties of Dðs; wÞ can be used to obtain non-vanishing results for special values and information on the asymptotic properties of Lðw; p; wn Þ. In [2] there is an excellent survey of such results, mostly due to Bump, Friedberg, Goldfeld and Ho¤stein. The
Forum Mathematicum – de Gruyter
Published: May 21, 2003
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