= Trivial zero of a Dirichlet L-function
= Trivial zeros of a Dirichlet L-function
{synonym}
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 $s=-a-2m$, $m=0,1,\ldots$. 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.
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