= Rankin–Selberg unfolding identity for a holomorphic Eisenstein series
{c}
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 sum>
$$
G_r(\tau)=\sum_{(m,n)\in\mathbb Z^2\setminus\{(0,0)\}}(m\tau+n)^{-r},
$$
whose constant <Fourier coefficient> is $2\zeta(r)$. With the <Petersson inner product> linear in its first argument and the elementary <Rankin–Selberg convolution> $L(f,g,s)=\sum_{n\ge1}a_n\overline{b_n}n^{-s}$, unfolding gives
$$
\langle f,gG_r\rangle=\frac{2\zeta(r)\Gamma(k-1)}{(4\pi)^{k-1}}L(f,g,k-1).
$$
The 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.
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