Nonvanishing of a nonprincipal Dirichlet L-function at one

For every nonprincipal Dirichlet character $\chi$,Multiply the Dirichlet L-functions over every character modulo $q$. The resulting Euler product has nonnegative coefficients. A zero at one would cancel the principal factor's pole and contradict the Landau theorem for a Dirichlet series with nonnegative coefficients.