Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

TL;DR

This paper numerically computes a special Dirichlet series related to Siegel modular forms of degree two, demonstrating its distinctness from spinor L-functions and developing a new numerical method for Petersson scalar products.

Contribution

It introduces a numerical approach to compute a Dirichlet series attached to Siegel modular forms and proves its independence from spinor L-functions.

Findings

The Dirichlet series shares the same functional equation as spinor L-functions.

The series is not a linear combination of spinor L-functions.

A new numerical method for Petersson scalar products is developed.

Abstract

The Rankin convolution type Dirichlet series $D_{F,G}(s)$ of Siegel modular forms $F$ and $G$ of degree two, which was introduced by Kohnen and the second author, is computed numerically for various $F$ and $G$. In particular, we prove that the series $D_{F,G}(s)$, which share the same functional equation and analytic behavior with the spinor $L$-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.