Rankin–Selberg unfolding identity for a holomorphic Eisenstein series

Let $f$ and $g$ be level-one cusp forms of respective weights $k$ and $l$, with $r=k-l$ an even integer greater than two. Use the unnormalized lattice Eisenstein sumwhose constant Fourier coefficient is $2\zeta (r)$. With the Petersson inner product linear in its first argument and the elementary Rankin–Selberg convolution $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$, unfolding givesThe sum over primitive bottom rows unfolds the fundamental domain to the translation strip. Integration in the real direction keeps equal Fourier modes, and ${\int }_0^{\infty }{e}^{-4\pi ny}{y}^{k-2}dy=\Gamma (k-1)/(4\pi n{)}^{k-1}$ supplies the factor. If instead one uses the normalized Eisenstein series $E_r=G_r/(2\zeta (r))$, the factor $2\zeta (r)$ is absent. The stated level and normalization are essential to the displayed formula.