If ρ : Gal(ℚac/ℚ))→GL2(ℂ) is a continuous odd irreducible representation wit nonsolvable image, then under certain local hypotheses we prove that ρ is the representation associated to a weight 1 modular form and hence that the L-function of ρ has an analytic continuation to the entire complex plane.
Kevin Buzzard. Mark Dickinson. Nick Shepherd-Barron. Richard Taylor. "On icosahedral Artin representations." Duke Math. J. 109 (2) 283 - 318, 15 Ausust 2001. https://doi.org/10.1215/S0012-7094-01-10922-8