Analytic continuation of a weight-k real-analytic Eisenstein series

For fixed $\tau$, Mellin transformation of a weighted Gaussian theta series expresses $G_k(\tau ,s)$ as a gamma factor times a Mellin integral. Splitting at one and applying Poisson summation to the small-time part continues it to all $s\in \mathbb{C}$; for positive even $k$, the reciprocal gamma factor cancels the apparent poles.