An explicit formula for Koecher-Maaß\ Dirichlet series for Eisenstein series of Klingen type.
Let \(f(Z)\) be a Siegel modular form of degree \(n\) and weight \(k\), with Fourier coefficients \(c_f(A)\) for semi-positive, half-integral \(n\times n\)-matrices \(A\). The Koecher-Maaß\ Dirichlet series of \(f\) is \[ L(f,s)= \sum_A {c_f(A)\over r(A,A)}\cdot\text{det}(A)^{-s}, \] where \(r(A, A)\) is the order of the orthogonal group of \(A\). Now, for \(0< r\leq n\leq k- r- 2\), let \(f\) be a Siegel modular form of degree \(r\) and weight \(k\), and let \([f]^n_r\) denote the Eisenstein series of Klingen type for \(f\), which is a Siegel modular form of degree \(n\) and weight \(k\). The authors raise the problem to find an explicit formula for \(L([f]^n_r, s)\) and to analyze the analytic and arithmetic properties of the Dirichlet series which appear in this formula. They offer an answer to this problem in the case when \(r= 1\), \(f\) is a cusp Hecke eigenform, and \(n\) is even. Naturally, the explicit formula in the answer is quite complicated. It involves, among other ingredients, a Dirichlet series \({\mathcal L}_{\pm 1}(f,\lambda,s)\) of a type which was introduced by \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, 175--198 (1981; Zbl 0468.10015)], and the standard zeta function \(\zeta^{\text{st}}(f, k- 1)\) of \(f\), which can be defined by means of Satake \(p\)-parameters of local Hecke algebras associated to \(f\). The research was inspired by previous work of \textit{S. Böcherer} and \textit{R. Schulze-Pillot} [Math. Ann. 288, No. 3, 361--348 (1990; Zbl 0688.10025); Math. Z. 209, No. 2, 273--288 (1992; Zbl 0773.11031); Math. Nachr. 169, 31--57 (1994; Zbl 0824.11028)].
scientific article; zbMATH DE number 2025727
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