If a Dirichlet $L$-function has no zeroes in $\Re(s) \gt c$, does its Euler product necessarily converge in $\Re(s) \gt c$?
So I know the proof that (conditional) convergence of the Euler product $\prod_p (1 - \chi(p)p^{-s})^{-1}$ for a nontrivial Dirichlet character $\chi$ in the region $\Re(s) > c$, $1/2 < c < 1$, is equivalent to the L-function $\sum_n \chi(n)n^{-s} = L(s, \chi)$ having no zeroes in this half-plane. Someone told me that the converse is also true, i.e. if the L-function $L(s, \chi)$ has no zeroes in such a half-plane, then the Euler product above for this L-function also converges conditionally in this range (obviously to the L-function itself). Does anyone know the technique to prove this, or can anyone direct me to a source that covers this exact result? Thank you!