  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 $0$.
