Real-analytic modular forms for $\Gamma_0(N)$ and their $L$-series

TL;DR

This paper introduces a new class of $L$-series associated with real-analytic modular forms for $ ext{Gamma}_0(N)$, establishing their functional equations and exploring examples like Eisenstein series and iterated integrals.

Contribution

It develops the theory of $L$-series for real-analytic modular forms of bi-weight $(r,s)$, including a converse theorem and explicit examples, extending prior work on holomorphic forms.

Findings

$L$-series satisfy functional equations

Includes examples like Eisenstein series and modular iterated integrals

Extends the theory to real-analytic forms of higher level

Abstract

We introduce an $L$-series associated to real-analytic modular forms which transform with weight $(r,s)\in\mathbb{Z}^2$ under $\Gamma_0(N)$. These $L$-series satisfy a functional equation and converse theorem. We also discuss examples of such forms, including real analytic Eisenstein series and modular iterated integrals of higher level. We focus on the $L$-series for a class of forms including Francis Brown's length-one modular iterated integrals $\mathcal{MI}^!_1$.