Real-analytic modular forms for ${\mathrm{\Gamma }}_0(N)$ and their $L$-series

We introduce an $L$-series associated to real-analytic modular forms which transform with weight $(r,s)\in {\mathbb{Z}}^2$ under ${\mathrm{\Gamma }}_0(N)$. 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 ${\mathcal{M}\mathcal{I}}_1^{!}$.