Weight-k real-analytic Eisenstein series ($E_{k,s}$)

For even $k$ and $\mathrm{Re}s>(2-k)/2$, defineIt is generally nonholomorphic and transforms with weight $k$.

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.

The absolute value of the summand indexed by a primitive bottom row $(c,d)$ isThe exponent exceeds two, so comparison with the lattice sum over $(c,d)\in {\mathbb{Z}}^2\setminus \{0\}$ proves absolute and locally uniform convergence.

If $f$ is a weight-$k$ modular form, thenis modular invariant. The factors transform by $j(\gamma ,\tau {)}^k$, ${\overline{j(\gamma ,\tau )}}^k$, and $|j(\gamma ,\tau ){|}^{-2k}$.