For a Hecke L-function, if all of the local eigenvalues are roots of unity, is it an Artin L-function?



A question on the Hecke L-function
Whatever your answer to my question in comment, the answer to the title question is yes. Take the case of the $L$-function attached to a modular form $f$, which we assume an eigenform for almost all Hecke operators (since otherwise there is no decomposition of the L-function into Euler product and the phrase "local eigenvalues" has no meaning.)


By Deligne, we know then that the local eigenvalues at $p$ for almost every $p$ are Weil numbers, more precisely they are algebraic numbers whose every embedding in $\mathbb C$ has absolute values $p^{(k-1)/2}$ where $k$ is the weight of your modular form (cf. the last corollary of this [link](http://www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf) for instance). If just one of those eigenvalue is a root of unity, this already forces $k=1$, in which case we know by Deligne-Serre that the Galois representation attached to $f$ has finite image, that is that $L(f,s)$ is an Artin $L$-function.


