Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod\_{j=1}^{r}\Gamma(\lambda\_{j}s+\mu\_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor $\gamma\_{F}$ of $F$. Can there exist a couple $(i,j)$ with $i\neq j$ such that $(\lambda\_{i},\mu\_{i})=(\lambda\_{j},\mu\_{j})$?  

Thanks in advance.



gamma-factor of a primitive element of the Selberg class
The answer is yes if we accept the Ramanujan conjecture for principal automorphic $L$-functions.


Indeed, the $L$-function of an even Maass form of Laplacian eigenvalue $1/4$ for some congruence subgroup of $\mathrm{SL}\_2(\mathbb{Z})$ has two gamma factors, each equal to $\Gamma(s/2)$. Moreover, by a result of Ram Murty (Selberg's conjectures and Artin L-functions. II, Current trends in mathematics and physics, 154-168, Narosa, New Delhi, 1995), the Ramanujan conjecture implies that this $L$-function is a primitive element of the Selberg class.


