Weight-k real-analytic Eisenstein series
= Weight-k real-analytic Eisenstein series
{title2=$E_{k,s}$}
For even $k$ and $\operatorname{Re}s>(2-k)/2$, define
$$
E_{k,s}(\tau)=
\sum_{\gamma\in\Gamma_\infty\backslash\Gamma(1)}
\operatorname{Im}(\gamma\tau)^s j(\gamma,\tau)^{-k}.
$$
It is generally nonholomorphic and transforms with weight $k$.