Calling '$L$-function' any automorphic $L$-function belonging to the Selberg class, what are the known $L$-functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the Rankin-Selberg convolution $L(s, F \otimes G)$ is provably (as of today, December 13 2016) an $L$-function of degree $dd'$?


An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This includes two important cases: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore the answer includes Hecke L-functions and L-functions of modular forms.

I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing. See the comments for more information.

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    $\begingroup$ I am fairly sure that the generalised Ramanujan conjecture is known in a bit more generality than that! See e.g. Clozel's article "Purity Reigns Supreme", imj-prg.fr/fa/bpFiles/Clozel2.pdf, which shows for instance that a self-dual cohomological cuspidal automorphic representation of $GL_n / \mathbf{Q}$ of conductor 1 must satisfy Ramanujan. $\endgroup$ – David Loeffler Dec 14 '16 at 9:46
  • $\begingroup$ @David Loeffler : would preprints.ihes.fr/2016/M/M-16-05.pdf provide other examples of L-functions satisfying the required property ? $\endgroup$ – Sylvain JULIEN Dec 14 '16 at 10:10
  • $\begingroup$ @DavidLoeffler You're right, I didn't mean to say that those were all the known cases. I'll fix it. Do you know of other cases where both Ramanujan and modularity of R-S are known? $\endgroup$ – Myshkin Dec 14 '16 at 11:47
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    $\begingroup$ What are you saying about GL(2) x GL(3)? Functoriality is known by Kim-Shahidi. $\endgroup$ – Kimball Dec 14 '16 at 14:55
  • $\begingroup$ I'd say It also includes the Rankin-Selberg convolution of those $\endgroup$ – reuns Dec 14 '16 at 15:30

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