(This answer is a community wiki version of a comment above by FC which answered the question.)
For any integer M, there exists a prime p such that chi_p(n) = (n/p) = 1 for all n = 1...M. This means that the Dirichlet series L(s,V chi_p) (for any representation V) "converges" in your sense to L(s,V). but they do not converge at s = 0. If V is trivial, then we are comparing zeta(0) = -1/2 with L(0,chi_p) which grows without bound by Brauer-Siegel. I think in this class of examples one does get convergence at the critical point.