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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.

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