Factorization in the extended Selberg class of L-functions associated to modular forms

Let $N\in\N$ and let $\chi$ be a Dirichlet character modulo $N$. Let $f$ be a modular form with respect to the group $\Gamma_0(N)$, multiplier $\chi$ and weight $k$. Let $F$ be the $L$-function associated with $f$ and normalized in such a way that $F(s)$ satisfies a functional equation where $s$ reflects in $1-s$. The modular forms $f$ for which $F$ belongs to the extended Selberg class $\Selberg^\sharp$ are characterized. For these forms the factorization of $F$ in primitive elements of $\Selberg^\sharp$ is enquired. In particular, it is proved that if $f$ is a cusp form and $F\in\Selberg^\sharp$ then $F$ is almost primitive (i.e., that if $F=PG$ is a factorization with $P,G\in\Selberg^\sharp$ and the degree of $P$ is $<2$ then $P$ is a Dirichlet polynomial). It is also proved that the conductor of the polynomial factor $P$ is bounded by $N$. If $f$ belongs to the space generated by newforms and $N\leq 4$ then $F$ is actually primitive (i.e., $P$ is a constant).