Real-analytic modular forms for Γ0(N) and their L-series

Abstract

We study real-analytic modular forms on congruence subgroups of the type ${\Gamma }_0\left(N\right)$. We examine their properties and discuss examples, such as real-analytic Eisenstein series and modular iterated integrals. We also associate an L-series to these forms and prove its functional equation. For the L-series of a special class of forms, which includes length-one modular iterated integrals, we establish a converse theorem.