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 Lfunctions with degree 1, and the Lfunctions 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. 

