Dirichlet L-Funktion for non-principal charakters at s=1

Question

It seems that in my introducction to analytic number theorie they provide an easy proof that if $\chi\ne \chi_1$, $L(s,\chi)$ is an entire function of s. Thus I do not understand this proof. It says: $$Res_{s=1}L(s,\chi)=\frac{1}{k}\sum_{r=1}^k\chi(r)=0\text{ if } \chi \ne \chi_1$$ using the charakter relation. I can easily understand how they get to this result. But I do not understand how does this proof that $L(s,\chi)$ is an entire function for non principal charakters, since $Res f=0$ do not imply that f is holomorphic at this point does it?