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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?

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I think the residue is $\langle f,\overline{f}\rangle$ (the Petersson inner product). Henri Darmon thought a course on L-functions last semester and there was a portion of the course devoted to the Rankin-Selberg method. Have a look at this: math.mcgill.ca/darmon/courses/nt/notes/lecture22.pdf (although it seems you don't have the same normalisation). –  M Turgeon Jul 18 '12 at 20:41
    
I think the residue is $\langle f,\bar{f}\rangle$ for the completed $L$-function, though obviously the two residues are related. Also, it seems that Darmon is handling the case where the character is trivial. –  Jim Jul 19 '12 at 0:00
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Are you sure that it has a pole at $s = 1$? I think the pole is actually at $s = k$. –  David Loeffler Jul 19 '12 at 7:08
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@David: He is using the "analytic" normalization so that the critical line is $\Re s =1/2$. –  Jim Jul 19 '12 at 18:04

1 Answer 1

If you write instead $$L(s,f\otimes f) = \sum_1^\infty \lambda_f^2(n)n^{-s},$$ then $$\operatorname{Res}_{s=1} L(s,f\otimes f)=Z(1,f)M/\varphi(M)$$ (equation (2.35) in Iwaniec-Luo-Sarnak: http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2000__91_/PMIHES_2000__91__55_0/PMIHES_2000__91__55_0.pdf).

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