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