Editing Dirichlet L-function (section)

==Zeros==
[[Image:Mplwp dirichlet beta.svg|thumb|right|300px|The Dirichlet ''L''-function ''L''(''s'', ''χ'') = 1 − 3<sup>−''s''</sup> + 5<sup>−''s''</sup> − 7<sup>−''s''</sup> + ⋅⋅⋅ (sometimes given the special name [[Dirichlet beta function]]), with trivial zeros at the negative odd integers]]

Let ''χ'' be a primitive character modulo ''q'', with ''q'' > 1.

There are no [[zero of a function|zeros]] of ''L''(''s'', ''χ'') with Re(''s'') > 1. For Re(''s'') < 0, there are zeros at certain negative [[integer]]s ''s'':
* If ''χ''(−1) = 1, the only zeros of ''L''(''s'', ''χ'') with Re(''s'') < 0 are simple zeros at −2, −4, −6, .... (There is also a zero at ''s'' = 0.) These correspond to the poles of <math>\textstyle \Gamma(\frac{s}{2})</math>.<ref name="DavenportCh9">{{harvnb|Davenport|2000|loc=chapter 9}}</ref>
* If ''χ''(−1) = −1, then the only zeros of ''L''(''s'', ''χ'') with Re(''s'') < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of <math>\textstyle \Gamma(\frac{s+1}{2})</math>.<ref name="DavenportCh9" />
These are called the trivial zeros.<ref name="MontgomeryVaughan333"/>

The remaining zeros lie in the critical strip 0 ≤ Re(''s'') ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(''s'') = 1/2. That is, if <math>L(\rho,\chi)=0</math> then <math>L(1-\overline{\rho},\chi)=0</math> too, because of the functional equation. If ''χ'' is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if ''χ'' is a complex character. The [[generalized Riemann hypothesis]] is the conjecture that all the non-trivial zeros lie on the critical line Re(''s'') = 1/2.<ref name="MontgomeryVaughan333" />

Up to the possible existence of a [[Siegel zero]], zero-free regions including and beyond the line Re(''s'') = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet ''L''-functions: for example, for ''χ'' a non-real character of modulus ''q'', we have

:<math> \beta < 1 - \frac{c}{\log\!\!\; \big(q(2+|\gamma|)\big)} \ </math>

for β + iγ a non-real zero.<ref>{{cite book |last=Montgomery |first=Hugh L. |author-link=Hugh Montgomery (mathematician) |title=Ten lectures on the interface between analytic number theory and harmonic analysis |series=Regional Conference Series in Mathematics |volume=84 |location=Providence, RI |publisher=[[American Mathematical Society]] |year=1994 |isbn=0-8218-0737-4 |zbl=0814.11001 |page=163}}</ref>