= Primitive real-analytic Eisenstein series
{title2=$E(\tau,s)=\frac12\sum_{\gcd(m,n)=1}\frac{y^s}{|m\tau+n|^{2s}}$}
For $\operatorname{Re}s>1$ this <Eisenstein series> also equals $\sum_{\gamma\in\Gamma_\infty\backslash SL_2(\mathbb Z)}(\operatorname{Im}\gamma\tau)^s$, where $\Gamma_\infty$ consists of translations and their negatives. It is <modular invariant>. The <nonholomorphic Eisenstein series> summed over all nonzero integer pairs instead equals $2\zeta(2s)E(\tau,s)$: separate each pair into a positive integer times a primitive pair. This normalization matters in <Rankin–Selberg unfolding>.
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