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In this paper, we study the Mellin transform of the weight 2 level $N$ polarharmonic Maass form $H_{N,z}^*(\tau)$, and analyze this (generalized)$L$-function as $\mathrm{Im}(z)\to \infty$. On the way, we also calculate theFourier expansion of $H_{N,z}^*(\tau)$ at arbitrary cusps of $\Gamma_0(N)$, andwe 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 $0$.

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