The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex plane? The Dedekind zeta function of a number field, which is a pure motive of weight 0, has a pole at $s=1$. Is this somehow the only case? (Meaning, I am just guessing, that for example if there are no Hodge classes then holomorphic continuation is expected?)

irreduciblethen the group you mention must be 0. That seems to suggest that the OP's guess is right. $\endgroup$ – Kapil Oct 22 '18 at 3:21cuspidalautomorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ does not have a pole at $s = 1$ (unless you consider the trivial rep of $\mathrm{GL}_1$ to be cuspidal); see my answer here: mathoverflow.net/questions/285135/… $\endgroup$ – Peter Humphries Oct 22 '18 at 9:25