Are there L-functions of degree 1 that aren't Hecke L-functions?

I don't know of any examples and I don't know of any results which prohibit them

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The answer is no. Kaczorowski and Perelli proved the classification of L-functions with degree 1, and the L-functions with that degree turn out to be Riemann's $\zeta$ and Dirichlet's $L(s+i\omega,\chi)$, $\chi$ primitive. You can find the proof in:
The question seems to allow implicitly for Hecke $L$-functions on all number fields to have "degree 1", while this answer is only allowing for Dirichlet $L$-functions with a vertical shift in $s$ (i.e., Hecke $L$-functions over $\mathbf Q$) to have degree 1, and that doesn't include all Hecke $L$-functions. So there is a mismatch in the meaning of the term "degree 1". The OP should clarify what is meant in the question by the term "degree 1". It is defined precisely in the context of the Selberg class, but that is probably more restrictive than what the OP has in mind. –  KConrad May 25 at 14:29