Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} \lambda_f(n)n^{(k-1)/2}e^{2\pi i n z}$$ for $\Im z >0$ where $\lambda_f(1)=1$ and, by Deligne's bound, $|\lambda_f(n)|\le d(n)$. Here $d(n)$ denotes the number of positive divisors of $n$. Define the Rankin-Selberg convolution $L$-function as $$ L(s,f\times \bar{f}) = \sum_{n=1}^\infty \frac{|\lambda_f(n)|^2}{n^s} $$ for $\Re s >1$. This $L$-function has a simple pole at $s=1$.
Questions:
(1) What is this residue?
(2) How does one compute this residue?