On a convolution series attached to a Siegel Hecke cusp form of degree 2

TL;DR

This paper proves the existence of a pole at s=1 for a convolution Dirichlet series associated with a degree 2 Siegel cusp form, and derives asymptotic formulas for eigenvalue sums and convergence properties of related zeta functions.

Contribution

It establishes the pole at s=1 for the convolution series and provides explicit asymptotics and convergence results for the associated eigenvalues and zeta functions.

Findings

Convolution Dirichlet series D_2(s) has a pole at s=1.

Asymptotic formula for partial sums of eigenvalues with explicit error.

Abscissa of absolute convergence of the spinor zeta function is s=1.

Abstract

We prove that the "naive" convolution Dirichlet series D_2(s) attached to a degree 2 Siegel Hecke cusp form F, has a pole at s=1. As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor zeta function attached to F is s = 1.