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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?

  • 0
    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: http://www.math.mcgill.ca/darmon/courses/nt/notes/lecture22.pdf (although it seems you don't have the same normalisation).2012-07-18
  • 0
    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.2012-07-19
  • 1
    Are you sure that it has a pole at $s = 1$? I think the pole is actually at $s = k$.2012-07-19
  • 2
    @David: He is using the "analytic" normalization so that the critical line is $\Re s =1/2$.2012-07-19
  • 0
    @MTurgeon, Could you send to me this notes, the link doesn't work with me2016-02-17
  • 0
    @Med Wow, that's old... Here's the new link: http://www.math.mcgill.ca/darmon/courses/11-12/nt/notes/lecture22.pdf2016-02-17
  • 0
    @MTurgeon, Thanks a lot2016-02-17

2 Answers 2