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

