= Analytic continuation of a weight-k real-analytic Eisenstein series
The primitive <weight-k real-analytic Eisenstein series> must be distinguished from the sum over all nonzero <integer> pairs. For positive even $k$, write $y=\operatorname{Im}\tau$ and define
$$
G_{k,s}(\tau)=\sum_{(m,n)\ne(0,0)}\frac{y^s}{(m\tau+n)^k|m\tau+n|^{2s}}
=2\zeta(2s+k)E_{k,s}(\tau)
$$
in the half-plane of absolute convergence. The equality follows by separating a nonzero pair into a positive <integer> times a primitive pair, with a factor two for simultaneous sign.
For fixed $\tau$, the all-pairs sum $G_{k,s}$ continues to an entire function of $s$. One way to see this is to introduce the weighted <theta function>
$$
\theta_k(t)=\sum_{\lambda\in\mathbb Z\tau+\mathbb Z}\left(\frac{\overline\lambda}{\sqrt y}\right)^k e^{-\pi t|\lambda|^2/y}.
$$
The zero term vanishes. The weight is a <harmonic polynomial>, so its Gaussian <Fourier transform> has zero constant mode as well. <Poisson summation> then gives rapid decay at both endpoints, up to powers of $t$, making its <Mellin transform> entire. Termwise integration in the initial half-plane yields
$$
\int_0^\infty\theta_k(t)t^{s+k-1}\,dt=\Gamma(s+k)\pi^{-(s+k)}y^{k/2}G_{k,s}(\tau).
$$
Multiplication by the entire reciprocal <gamma function> gives the claimed continuation. The primitive series instead continues meromorphically as $E_{k,s}=G_{k,s}/(2\zeta(2s+k))$; it must not inherit an unqualified entire-function claim after division by the <Riemann zeta function>.
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