Selberg class and unique factorization Hello,
could someone tell me which are the elements of the Selberg class for which unique factorization into primitive elements has been proven up to now?
Thank you in advance.
 A: Caveat: Due to my limited knowledge of the subject, this is only a (very) partial answer. 
As far as I understand, very little is known unconditionally about unique factorization in the Selberg class $\mathcal S$. If one is willing to work in the larger extended Selberg class $\mathcal S^\sharp$ introduced by Kaczorowski and Perelli in 
J. Kaczorowski, A. Perelli, On the structure of the Selberg class, I: $0\leq d\leq1$, Acta Math. 182 (1999), 207-241
then an unconditional result has been proved by Kaczorowski, Molteni and Perelli in 
J. Kaczorowski, G. Molteni, A. Perelli, Unique factorization results for semigroups of $L$-functions, Math. Ann. 341 (2008), 517-527.
Namely, it is shown that the subsemigroup $\mathcal S_0^\sharp$ of $\mathcal S^\sharp$ consisting of degree $0$ functions has unique factorization (here the degree of a function $F\in\mathcal S^\sharp$ is a suitable real number defined in terms of the $\Gamma$-factors appearing in the functional equation for $F$). The same conclusion is true for the semigroup $\mathcal G_1^\sharp$ generated by the functions $F\in\mathcal S^\sharp$ of degree $0$ or $1$. 
It is perhaps worth remarking that $\mathcal S^\sharp$ is made up of those functions which satisfy the same axioms as those enjoyed by the functions in $\mathcal S$ with the exceptions of the Ramanujan hypothesis and the Euler product property. Moreover, it is known that the Selberg Orthonormality Conjecture does not hold in $\mathcal S^\sharp$ (whereas it is conjectured to hold in the smaller class $\mathcal S$).
Finally, essentially as a corollary of the above-mentioned result, (unique) factorization properties of $L$-functions associated with (holomorphic) modular forms are studied in
G. Molteni, Factorization in the extended Selberg class of $L$-functions associated with holomorphic modular forms, Math. Nachr. 282 (2009), 232-242.
