Rankin–Selberg convolution ($L(f,g,s)$)

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 $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

For normalized level-one Hecke eigenforms, let ${\alpha }_p+{\beta }_p=a_p$, ${\alpha }_p{\beta }_p=p^{k-1}$, and similarly ${\gamma }_p+{\delta }_p=b_p$, ${\gamma }_p{\delta }_p=p^{k-1}$. The Hecke multiplication relations give $a_{p^r}=({\alpha }_p^{r+1}-{\beta }_p^{r+1})/({\alpha }_p-{\beta }_p)$, with the repeated-root case obtained by continuity. Multiplying the two expressions and summing four geometric series proves the displayed local factor. Multiplicativity gives an Euler product; multiplying by $\zeta (2s-2k+2)$ cancels its local numerator and produces the degree-four convolution factors.

For two level-one weight-$k$ cusp forms, $F(s)=\sum a_n{\overline{b}}_n{n}^{-s}$. Unfolding against the nonholomorphic lattice sum normalized by ${\pi }^{-w}\Gamma (w)$ gives the displayed completion as the integral of $f\overline{g}{y}^k$ against that Eisenstein series. Its functional equation is $\mathcal{R}(s)=\mathcal{R}(2k-1-s)$, and its only possible simple poles are at $k-1,k$. The raw series is obtained by dividing out the gamma and zeta factors, so it does not inherit the same unqualified pole statement; zeros of the denominator zeta factor can yield additional possible poles. The raw series is regular at $k-1$ and can have a simple pole at $k$.

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