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

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 $0$.