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Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$.

What is the complete $L$-function of the Rankin-Selberg product $L(s,f\times \tilde f)$? I am having trouble writing down the local part at $p|N$. I can't find good references either.

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See here for the answer (in particular, it's in section 1 and 2). Basically, the answer isn't pretty - it depends on the local components of the automorphic representation $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ associated to the Maass form $f$.

So if $p \nmid N$, then the local component $\pi_p$ of $\pi$ is a principal series representation $\omega_{p,1} \boxplus \omega_{p,2}$, where $\omega_{p,1}, \omega_{p,2}$ are unramified characters of $\mathbb{Q}_p^{\times}$. We have that \[L_p(s,\pi_p) = \frac{1}{(1 - \omega_{p,1}(p) p^{-s})(1 - \omega_{p,2}(p) p^{-s})}\] where $\omega_{p,1}(p) + \omega_{p,2}(p) = \lambda_f(p)$, the Hecke eigenvalue of $f$ at $p$, and $\omega_{p,1}(p) \omega_{p,2}(p) = \chi(p)$. Then the local Rankin-Selberg $L$-function is \[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{(1 - |\omega_{p,1}(p)|^2 p^{-s})(1 - \omega_{p,1}(p) \overline{\omega_{p,2}(p)} p^{-s})(1 - \overline{\omega_{p,1}(p)} \omega_{p,2}(p) p^{-s})(1 - |\omega_{p,2}(p)|^2 p^{-s})}.\]

If $p \mid N$ and $\pi_p$ is a principal series representation $\omega_{p,1} \boxplus \omega_{p,2}$, where at least one of $\omega_{p,1}, \omega_{p,2}$ is ramified, then the same result holds as for the case $p \nmid N$ with the minor change that if $|\omega_{p,1}|^2$ is a ramified character of $\mathbb{Q}_p^{\times}$, we replace $|\omega_{p,1}(p)|^2$ with $0$, and similarly if $\omega_{p,1} \overline{\omega_{p,2}}$, $\overline{\omega_{p,1}} \omega_{p,2}$, or $|\omega_{p,2}|^2$ is ramified (see Jeremy's comment below). Note that if both $\omega_{p,1}$ and $\omega_{p,2}$ are ramified then $p^2 \mid N$. This is part of Proposition 1.4 of the linked paper.

If $p \mid N$ and the local component $\pi_p$ of the automorphic representation is supercuspidal (which can only happen if $p^2 \mid N$), then $\lambda_f(p) = 0$ and \[L_p(s,\pi_p) = 1,\] and either \[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{1 - p^{-s}}\] or \[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{1 - p^{-2s}}.\] The former occurs if $\pi_p \ncong \pi_p \otimes \eta_p$, where $\eta_p$ is the unramified quadratic character of $\mathbb{Q}_p^{\times}$, while the latter occurs if $\pi_p \cong \pi_p \otimes \eta_p$. This is Corollary 1.3 of the linked paper.

If $p \mid N$ and the local component $\pi_p$ is a Steinberg representation $\omega_p \mathrm{St}_p$, where $\omega_p$ is a unitary character of $\mathbb{Q}_p^{\times}$, then $\widetilde{\pi_p} = \omega_p^{-1} \mathrm{St}_p$. We have that \[L_p(s,\pi_p) = L_p\left(s + \frac{1}{2}, \omega_p\right)\] and \[L_p(s,\pi_p \times \widetilde{\pi_p}) = \zeta_p(s + 1) \zeta_p(s) = \frac{1}{(1 - p^{-s-1})(1 - p^{-s})}.\] This is part of Proposition 1.4. Note that if $\omega_p$ is unramified, we must have that $p \parallel N$, $\lambda_f(p) = \omega_p(p) p^{-1/2}$, and so \[L_p(s,\pi_p) = \frac{1}{1 - \omega_p(p) p^{-s - 1/2}},\] If $\omega_p$ is ramified, then $p^2 \mid N$, $\lambda_f(p) = 0$, and \[L_p(s,\pi_p) = 1.\] In particular, if $N$ is squarefree and $\chi$ is the principal character, then for $p \mid N$, $\pi_p$ must be $\omega_p \mathrm{St}_p$ with $\omega_p$ the trivial character, so $\lambda_f(p) = p^{-1/2}$.

Finally, the archimedean $L$-function is \[L_{\infty}(s,\pi_{\infty}) = \pi^{-s - \kappa} \Gamma\left(\frac{s + i\mu + \kappa}{2}\right) \Gamma\left(\frac{s - i\mu + \kappa}{2}\right),\] where $\kappa = 0$ if $f$ is even and $1$ if $f$ is odd, and \[L_{\infty}(s,\pi_{\infty} \times \widetilde{\pi_{\infty}}) = \pi^{-2s} \Gamma\left(\frac{s + 2i\mu}{2}\right) \Gamma\left(\frac{s}{2}\right)^2 \Gamma\left(\frac{s - 2i\mu}{2}\right).\]

The completed $L$-function $\Lambda(s,f \times \widetilde{f})$ is then defined to be \[\Lambda(s,f \times \widetilde{f}) = L_{\infty}(s,\pi_{\infty} \times \widetilde{\pi_{\infty}}) \prod_p L_p(s,\pi_p \times \widetilde{\pi_p}).\] Theorem 2.2 of the linked paper states that this extends meromorphically to the entire complex plane with only simple poles at $s = 0$ and $s = 1$.

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  • $\begingroup$ A minor clarification: on the subject of how we should interpret $|\omega_{1,p}(p)|^{2}$, I think we should let $\chi = \omega_{1,p} \overline{\omega_{1,p}}$ and then if interpret $|\omega_{1,p}(p)|^{2}$ as being equal to $\chi(p)$, if $\chi$ is unramified. (Note that we can have $\chi$ be unramified even if $\omega_{1,p}$ is.) $\endgroup$ – Jeremy Rouse Mar 12 '15 at 12:21
  • $\begingroup$ @JeremyRouse, yes, this is completely correct! $\endgroup$ – Peter Humphries Mar 12 '15 at 13:06
  • $\begingroup$ Very good summary. Note that for $\pi$ coming from a Maass form of odd weight, the archimedean factors of $L(s,\pi_\infty)$ are slightly different. Good references are Proposition 8.1 in Duke-Friedlander-Iwaniec: The subconvexity problem for Artin L-functions (Inventiones, 2002) and Section 11.14 in Goldfeld-Hundley: Automorphic Representations and L-Functions for the General Linear Group, Volume 1 (Cambridge University Press, 2011). $\endgroup$ – GH from MO Mar 16 '15 at 14:38
  • $\begingroup$ @GHfromMO, thanks, I think it should be all correct now (there was a missing $\pi^{-\kappa}$ term). Also I'm not sure why this was just downvoted... $\endgroup$ – Peter Humphries Mar 17 '15 at 18:11
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    $\begingroup$ To add to this 3 year old comment chain, the Proposition 8.1 of Duke-Friedlander-Iwaniec has a slight error which affects if the $\kappa = 1$ shift in the gamma factor occurs with $\mu$ or $-\mu$. I corrected it in Section 12 of arxiv.org/pdf/1710.03624.pdf $\endgroup$ – Matt Young Mar 19 '18 at 21:03

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