Petersson inner product

For weight-$k$ cusp forms on $\Gamma$, the Petersson inner product isCusp decay makes the integral convergent.

The Rankin–Selberg method represents Dirichlet series built from automorphic forms as integrals against Eisenstein series and studies them by unfolding those integrals.

For cusp forms $f=\sum a_n{q}^n$ and $g=\sum b_n{q}^n$, their Rankin–Selberg convolution in the elementary normalization is $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

If $f$ and $g$ have respective weights $k$ and $l$, and $r=k-l>2$, unfolding the weight-$r$ Eisenstein series gives