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The identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1979), Part 2, 63-86 …

In what sense is the Weyl law different for congruence subgroups and cocompact groups? At any rate, the Jacquet-Langlands correspondence is not a bijection between the two cuspidal spectra. More pre …

I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two …

The Langlands functoriality conjecture implies that automorphic $L$-functions belong to the Selberg class, but not the other way (i.e. the other direction is not known to follow from this conjecture). …

Let $Q=Q(\pi,it)$ be the analytic conductor of $\pi\otimes|\det|^{it}$. For $\pi$ not self-dual, the method of de la Vallée-Poussin yields the zero-free region $\sigma>1-c/\log Q$ for some effective c …

I think this is widely open. Flicker has a conditional result under certain cases of the Artin conjecture for Artin $L$-functions, see the Theorem on Page 232 of Pacific J. Math. 154 (1992). In partic …

This is an elaboration of Lucia's comment. Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following …

Yes, we have $L(s,\pi_p\otimes\chi_p)=1$ if the conductor of $\chi_p$ exceeds the conductor of $\pi_p$. By the inductive nature of the local $L$-function (see Jacquet: Principal $L$-functions of the …

I think your questions are answered in Asai's original paper (Math. Ann. 226 (1977), 81-94). Theorem 1 (on page 86) describes holomorphicity, poles, and the functional equation. Theorem 2 (on page 87) …