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A function on the complex upper half-plane is modular invariant if it is unchanged by the fractional-linear action of the modular group. This does not imply holomorphicity: a nonholomorphic Eisenstein series is modular invariant, whereas a modular function additionally satisfies meromorphicity and cusp conditions.

Work initially with $\mathrm{Re}s>1$, where the nonholomorphic Eisenstein series and its differentiated series converge locally absolutely. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, the identitiesshow that each summand at $\gamma \tau$ becomes the summand indexed by $(ma+nc,mb+nd)$ at $\tau$. Reindexing the nonzero integer pairs therefore provesHere $y^s$ and the denominator powers use logarithms of positive real numbers, so there is no ambiguity for complex $s$.

For the nonnegative Laplace-Beltrami operator of the hyperbolic plane, $\Delta y^s=-y^2{\partial }_y^2{y}^s=s(1-s)y^s$. If $(m,n)=r(c,d)$ with $(c,d)$ primitive and $r>0$, choose a matrix in $S{L}_2(\mathbb{Z})$ with bottom row $(c,d)$. The corresponding summand isThe Laplace-Beltrami operator commutes with hyperbolic isometries, so this summand has eigenvalue $s(1-s)$. This also covers $m=0$, or follows there directly. Termwise differentiation now givesThe equation describes an automorphic eigenfunction; it does not assert that this noncuspidal function belongs to the square-integrable discrete spectrum.

Write $\tau =x+iy$ and $\sigma =\mathrm{Re}s>1$. The positive-definite quadratic formis bounded below by $C_{\tau }(m^2+n^2)$ for some $C_{\tau }>0$. Henceand the last two-dimensional lattice sum converges for $\sigma >1$. Thus the nonholomorphic Eisenstein series converges absolutely.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,andThe map $(m,n)\mapsto (ma+nc,mb+nd)$ permutes ${\mathbb{Z}}^2\setminus \{0\}$, so absolute convergence permits reindexing and gives $G(\gamma \tau ,s)=G(\tau ,s)$. Hence $G(\tau ,s)\in W_0$.

For $\mathrm{Re}s>1$ this Eisenstein series also equals ${\sum }_{\gamma \in {\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})}(\mathrm{Im}\gamma \tau {)}^s$, 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.