Testing the functional equations of a high-degree Euler product

TL;DR

This paper investigates the functional equations of high-degree L-functions related to Siegel modular forms, combining theoretical analysis with numerical testing to advance understanding of their properties.

Contribution

It introduces a precise framework for testing conjectured functional equations of high-degree L-functions and applies it to specific cases of degree 10, 14, and 16.

Findings

Representation theoretic calculations for degrees 10, 14, 16 L-functions

Development of a framework for testing functional equations

Numerical verification of the degree 10 adjoint L-function's functional equation

Abstract

We study the L-functions associated to Siegel modular forms (equivalently, automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation for an L-function, and we apply this to the degree 10 adjoint L-function associated to a Siegel modular form.