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.

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 the Riemann zeta function and Dirichlet L-functions with primitive characters and their twists. You can find the original proof in:

• J, Kaczorowski & A. Perelli, On the structure of the Selberg class, I (1999)