Petersson scalar products and L-functions arising from modular forms

TL;DR

This paper explores the analytic continuation and functional equations of Dirichlet series derived from modular forms, with applications to quadratic forms, expanding understanding of their properties beyond classical cases.

Contribution

It introduces new methods for analyzing Dirichlet series from modular forms with differing weights or characters, and applies these to quadratic forms.

Findings

Established analytic continuation for a broad class of Dirichlet series

Derived functional equations linking these series to modular forms

Applied results to problems involving quadratic forms

Abstract

Analytic continuation and functional equation of a Dirichlet series constructed from two (not necessarily cuspidal) holomorphic modular forms is discussed, where either weights of the modular forms or characters are not necessarily equal to each other. The applications to quadratic forms are also given.