Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions. 


My question is: does the unique factorization imply a weakened form of Ramanujan conjecture?
Thanks in advance.



Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan conjecture?
This seems unlikely to me. Philosophically unique factorization means something like the following: if $R$ is a (say completely reducible) representation of a group $G$ then the trace of $R$ determines the irreducible constituent representations. This will be true if the characters satisfy an orthogonality relation. Selberg's orthogonality conjecture is roughly that the trace of $\pi \times \check \pi$ is small if $\pi \not \simeq \check \pi$, which is sufficient to get the unique factorization. In fact one can get by with a weaker version of this "small", which is what Liu and Ye do to prove "factorizations are unique" (though they don't prove that factorizations exist, which would be a big deal).


On the other hand, if you just start off knowing unique factorization (i.e., the trace of $R$ determines the irreducible constituents), this doesn't a priori imply orthogonality. (If it did, one would presumably be able to get Selberg's orthogonality conjecture.) It doesn't even seem to imply that the traces of $\pi \times \check \pi$ ($\pi \not \simeq \check \pi$) should be small. All it seems to imply is something like the irreducible characters of $G$ are linearly independent (so if $G$ were a finite group, they could be a priori just some random $\mathbb Z$-basis for the space of class functions (well, such that any group character is a non-negative integral linear combination of them)).


