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.
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asked Jul 1, 2015 at 13:45
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1$\begingroup$ This doesn't quite answer the question, but in some lucky cases one can explicitly describe the possible gamma factors: for instance, if $\chi \mod q$ is a primitive Dirichlet character, all gamma factors of $L(s,\chi)$ look like $$ Q^s \prod_i\Gamma \left(\frac{s}{2m_i}+\frac{2a_i+(1+\chi(-1))/2}{2m_i}\right),$$ where $(a_i,m_i)_i$ is any exact covering system and $\displaystyle Q=\sqrt{\frac{q}{\pi}\prod_i m_i^{1/m_i}}$. $\endgroup$– user41593Commented Jul 1, 2015 at 15:08
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$\begingroup$ I sent you an e-mail with the considered article as an attached document. $\endgroup$– Sylvain JULIENCommented Jul 1, 2015 at 17:10
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1 Answer
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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.
answered Jul 1, 2015 at 15:20
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2$\begingroup$ Those are not known to be in the Selberg class, let alone be primitive. $\endgroup$– MyshkinCommented Jul 1, 2015 at 15:55
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$\begingroup$ @Myshkin: You are right, I will update my text accordingly. $\endgroup$ Commented Jul 1, 2015 at 16:17