The zeros of an L-function $L(s)$ that occur at
poles of its gamma factors are **trivial zeros**.

The presence of trivial zeros is required by the functional equation. In its analytic normalization, the L-function $L(s)$ has no zeros or poles in the right half-plane $\Re(s)>1$, but the gamma factors that appear in the functional equation have poles. Since the completed L-function has no poles (except possibly from Riemann zeta-function factors), the L-function must have zeros at poles of the gamma factors.

For arithmetic L-functions, the poles are at certain negative integers.

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- Review status: beta
- Last edited by David Farmer on 2019-05-14 07:36:16

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- 2019-05-14 07:36:16 by David Farmer
- 2019-05-14 06:51:07 by David Farmer
- 2019-05-14 06:38:03 by Andrew Sutherland

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