 Trivial zero of a Dirichlet L-function

For a primitive nonprincipal Dirichlet character with parity $a$, the gamma function poles in its completed Dirichlet L-function force zeros of $L(s,\chi )$ at $s=-a-2m$, $m=0,1,\dots$. The functional equation and nonvanishing of nonprincipal Dirichlet L-functions at one show that these zeros are simple and cancel the poles, rather than remaining zeros of the completion. For the Dirichlet beta function, the odd-parity examples are $\beta (-1)=\beta (-3)=\cdots =0$. The primitive principal conductor-one case is different: its zeta trivial zeros start at $-2$ and the completion has poles at zero and one.