Rankin–Selberg integral for holomorphic cusp forms ($I(f,g,w)={\int }_{\Gamma \backslash \mathbb{H}}f\overline{g}E(\tau ,w)y^{k}d\mu$)

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, and integrating the Fourier series in $x$. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.

For real $w>1$, the finite Rankin–Selberg integral $I(f,f,w)$ has a nonnegative integrand. Tonelli theorem unfolds it without an a priori bound on the Dirichlet series. Integration over $x$ gives ${\int }_0^1|f(x+iy){|}^{2}dx={\sum }_{n\ge 1}|a_n{|}^2{e}^{-4\pi ny}$. A second application of Tonelli theorem gives the finite sum in the displayed identity with $s=w+k-1$. Positivity, rather than analytic continuation alone, proves absolute convergence.

By positive unfolding of a cusp-form square, $\sum |a_n{|}^2{n}^{-\beta }<\infty$ for $\beta >k$. Choose $k<\beta <2\sigma -1$. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by