I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under Rankin-Selberg convolution or not.


Many thanks in advance.



Consequences of Langlands functoriality conjecture
The Langlands functoriality conjecture implies that automorphic $L$-functions belong to the Selberg class, but not the other way (i.e. the other direction is not known to follow from this conjecture). Regarding to Rankin-Selberg convolutions, I don't think that this operation has been defined precisely for the Selberg class. There is a subtlety at the ramified primes: one usually refers to the algebraic classification of the underlying local representations, which is itself part of the Langlands program.


