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 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.