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

TL;DR

This paper investigates the Mellin transform and $L$-function of a weight 2 polar harmonic Maass form, providing Fourier expansions, functional equations, and local factorization, enriching the understanding of these automorphic objects.

Contribution

It introduces the analysis of the Mellin transform of the weight 2 polar harmonic Maass form and derives its functional equation and local factors, which are new contributions.

Findings

Derived the Fourier expansion at arbitrary cusps.

Established the functional equation of the $L$-function.

Factored the $L$-function into local components.

Abstract

In this paper, we study the Mellin transform of the weight 2 level $N$ polar harmonic Maass form $H_{N,z}^*(\tau)$, and analyze this (generalized) $L$-function as $\mathrm{Im}(z)\to \infty$. On the way, we also calculate the Fourier expansion of $H_{N,z}^*(\tau)$ 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 $0$.