Nonholomorphic Eisenstein series ($G(\tau ,s)$)

For $\mathrm{Re}s>1$,is an absolutely convergent modular-invariant function.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has