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 $dd'$?



Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function
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.


