the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.



Consequences of the degree conjecture
I think there are no direct consequences of the conjecture, but it seems like the natural first step in solving the bigger orthonormality conjecture (and/or related issues like unique factorization) which in turn imply a major open problems in number theory, the Artin conjecture (Dedekind conjecture, in the case of unique factorization).


You can find a proof of both implications in:


* M. Ram Murty, [Selberg's Conjectures and Artin L-functions](http://arxiv.org/abs/math/9407219) (1994)


And more information on the relationship between degrees and orthonormality in Kaczorowski and Perelli's series of articles on the Selberg class.


