An explicit formula for the Koecher-Maaß\ Dirichlet series for the Ikeda lifting
Let \(F\) be a Siegel modular form of degree \(n\) and weight \(k\), with Fourier coefficients \(a_F(A)\) for half-integral matrices \(A\geq 0\). Its Koecher-Maaß\ Dirichlet series is \[ L(F, s)= \sum_A {a_F(A)\over e(A)}(\text{det\,}A)^{-s}, \] where \(e(A)\) is the order of the orthogonal group of \(A\), and \(A\) runs over representatives of \(\text{SL}_n(\mathbb{Z})\)-equivalence classes of matrices \(A> 0\). If \(F\) is a lifting of a modular form \(f\) of degree \(m< n\), the authors pose the problem to express \(L(F, s)\) in terms of Dirichlet series attached to \(f\). Recently [J. Number Theory 102, No. 2, 223--256 (2003; Zbl 1053.11046)] they gave an answer when \(F\) is the Eisenstein series of Klingen type of a Hecke eigenform for \(\text{SL}_2(\mathbb{Z})\). In the present paper they consider the case when \(F= I(f)^n\) is the lifting [\textit{T. Ikeda}, Ann. Math. (2) 154, No. 3, 641--681 (2001; Zbl 0998.11023)] of a cuspidal Hecke eigenform \(f\) of weight \(2k- n\) for \(\text{SL}_2(\mathbb{Z})\). They are able to express \(L(I(f)^n, s)\), for even \(n\) and \(k\), by the Hecke \(L\)-function associated to \(f\) and by a convolution product \(L(\widetilde f,E_{(n+1)/2}, s)\) of the modular form \(f\) in the Kohnen plus-space corresponding to \(f\) and the Cohen Eisenstein series \(E_{(n+1)/2}\). As corollaries, one gets a meromorphic continuation and a functional equation for \(L(\widetilde f,E_{(n+1)/2}, s)\) and the algebraicity of special values of \(L(I(f)^n, s)\).
scientific article; zbMATH DE number 2139426
scientific article