The Completed $L$-function attached to the Weight 2 Polar Harmonic Maass Form $H_{N,z}^{\ast }(\tau )$

Abstract

In this paper, we study the Mellin transform of the weight 2 level $N$ polar harmonic Maass form $H_{N,z}^{\ast }(\tau )$, and analyze this (generalized) $L$-function as $\mathrm{Im}(z)\to \infty$. On the way, we also calculate the Fourier expansion of $H_{N,z}^{\ast }(\tau )$ at arbitrary cusps of ${\Gamma }_0(N)$, and we give a functional equation and factorization into local factors of the $L$-function for the weight 2 level $N$ Eisenstein series at the cusps $i\infty$ and $0$.

32 pages, no figures or tables