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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:

J, Kaczorowski & A. Perelli, "On the structure of the Selberg class, I: 0≤d≤1" (1999).

and most surveys on Selberg class.

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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
    
And yet I think it is standard to define the degree as the number of Gamma factors (normalized with an s/2) appearing in the completed L-function. In this sense for example the Dedekind zeta function of an abelian extension are degree n > 1, unless they correspond to \zeta(s). I think the OP should definitely rephrase his question if he meant something else (I actually fail to see the implicit version of the question that you mention) –  guest007 May 25 at 21:17

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