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Suppose $F$ is a primitive function of the Selberg class and $G$ a Dirichlet L-function. Is the Rankin Selberg convolution $F\otimes G$ itself primitive? This is true if $F$ is itself a Dirichlet L-function, since it has degree $1$ and so has the resulting L-function. But what do we know for primitive L-functions of degree greater than $1$? Thanks in advance.

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This is an important open problem. It is sometimes called Selberg's Twist Conjecture, and essentially no progress has been made on this problem, even for degree $2$ $L$-functions.

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  • $\begingroup$ you mean the twist up to the "equal for almost every prime power" equivalence ? (since $L(s,\chi) \otimes L(s,\overline{\chi}) = L(s,|\chi|^2)$ is not even in the Selberg class when $\chi$ is a Dirichlet character) $\endgroup$ – reuns Aug 19 '16 at 9:51
  • $\begingroup$ and my implicit question is how do you show there is always an element $H$ of the Selberg class such that $F \otimes G \cong H$ ? $\endgroup$ – reuns Aug 19 '16 at 9:54

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