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Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does the primitivity of $F\otimes G$ imply the primitivity of both $F$ and $G$?

Edit: perhaps I should add some motivation. I learnt that an irreducible representation of the direct product of two finite groups $G_{1}$ and $G_{2}$ is the tensor product of irreducible representations of the respective groups, so drawing and analogy between irreducible representation of a finite group and primitive L-function, I came to think about this question.

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  • $\begingroup$ Probably not. Among Artin L-functions, the primitive ones are the ones associated to irreducible representations, but tensor product of irreps need not be an irrep. The question is whether those satisfy your definition of L-rig, but conjecturally they surely should. The converse implication appears trivial to me. $\endgroup$
    – Wojowu
    Commented Dec 4, 2021 at 12:21
  • $\begingroup$ Didn't Ram Murty prove in 1994 that relationship between primitivity and irreducibility for Artin L-functions under Selberg orthogonality conjecture? Is it now proven unconditionally? $\endgroup$
    – Sylvain JULIEN
    Commented Dec 4, 2021 at 14:33
  • $\begingroup$ I'm not sure, probably not in general but it might be known in enough cases to answer the question. Take my comment more as an indication that one should expect negative answer, because of some standard conjectures. $\endgroup$
    – Wojowu
    Commented Dec 4, 2021 at 16:41

1 Answer 1

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Let $L(s,F)$ be the $L$-function of a self-dual $\mathrm{GL}(2)$ holomorphic cuspidal newform without complex multiplication and with trivial nebentypus. Let $G=\mathrm{Sym}^2 F$ be the symmetric square lift of $F$. Then $L(s,F)$ is a primitive $\mathrm{GL}(2)$ $L$-function (due to Hecke) and $L(s,G)$ is a primitive $\mathrm{GL}(3)$ $L$-function (due to Gelbart and Jacquet), but

$$L(s,F\otimes G) = L(s,F)L(s,\mathrm{Sym}^3 F),$$

which is not primitive. Simpler and more complicated examples abound; this seemed like a nice example from the middle ground.

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