Residue of Rankin Selberg L-function Let $f$ be a normalized holomorphic cusp form with weight $k$, level $N$. The Fourier expansion of $f$ can be written as
\begin{align*}
f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{(k-1)/2} e^{2\pi inz}
\end{align*}
The Rankin-Selberg convolution is defined as $L(f\times f) = \sum_{n=1}^{\infty}\frac{\lambda_f(n)^2}{n^s}$ for $\Re s>1$. How to calculate the residue at $s=1$ i.e. $\mathrm{Res}_{s=1}L(f\times f)$?
Thanks in advance!

The integral representation of $L(s,f\times f)$ (with suitable normalization) was shown (by Rankin and by Selberg, in 1939), to be obtained by integrating $|f|^2$ against the Eisenstein series $E_s$. If we have the normalizations set up appropriately, then the residue at the first pole is the residue of that integral, which is the integral of $|f|^2$ against the residue of $E_s$ at $s=1$ (a constant, etc.) So, up to normalization (which is not so hard to nail down), that first residue is the integral of $|f|^2$...
In addition to the two papers from 1939, by now most introductions to modular forms include this computation. I treat the simplest case in a small essay http://www.math.umn.edu/~garrett/m/v/basic_rankin_selberg.pdf
