In Problems in Algebraic Number Theory from M. Ram Murty, he mentions on page 148 how one show non-vanishing at 1 for a certain $L$-series. Here, he works with a generalization of Dirichlet $L$-series (a special case of Hecke $L$-series), that is $L$ functions associated to character of the ideal class group of a number field $K$. When he comes to the point where we specialize to a real character (as in the proof for the usual Dirichlet characters), he mentions there that it is possible to find an analytic continuation to $\Re(s) \geq 1/2$, but he does not provide a reference for this fact.
Does anyone know if there is a relatively simple proof for this? That is $L(s, \chi)$ admits an analytic continuation to $\Re(s) \geq 1/2$ when $\chi$ is a real character of the ideal class group?
I know that there is a strong theorem due to Hecke that provides continuation to $\mathbb{C}$ due, but I would like to avoid using it. Moreover, I would like to know if one can prove a similar result when the ideal class group is replaced by an arbitrary ray class group modulo $\mathfrak{m}$.
Using the class formula, and a "class formula" for ray class group, I have been able to find an analytic continuation for $\Re(s) > 1 - \frac{1}{[K : \mathbb{Q}]}$ for any character of a ray class modulo $\mathfrak{m}$, but this is not sufficient for my purposes here.
Thanks in advance for your answers!
