Editing Dirichlet L-function (section)

==Functional equation==

Dirichlet ''L''-functions satisfy a [[functional equation]], which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of <math>L(s,\chi)</math> to the value of <math>L(1-s, \overline{\chi})</math>. Let ''χ'' be a primitive character modulo ''q'', where ''q'' > 1. One way to express the functional equation is:<ref name="MontgomeryVaughan333" />
:<math>L(s,\chi) = W(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin \left( \frac{\pi}{2} (s + \delta) \right)  \Gamma(1-s) L(1-s, \overline{\chi}).</math>
In this equation, Γ denotes the [[gamma function]]; 
:<math>\chi(-1)=(-1)^{\delta}</math> ; and
:<math>W(\chi) = \frac{\tau(\chi)}{i^{\delta} \sqrt{q}}</math>
where ''τ''{{hairsp}}({{hairsp}}''χ'') is a [[Gauss sum]]:
:<math>\tau(\chi) = \sum_{a=1}^q \chi(a)\exp(2\pi ia/q).</math>
It is a property of Gauss sums that |''τ''{{hairsp}}({{hairsp}}''χ''){{hairsp}}| = ''q''<sup>1/2</sup>, so |''W''{{hairsp}}({{hairsp}}''χ''){{hairsp}}| = 1.<ref name="MontgomeryVaughan332">{{harvnb|Montgomery|Vaughan|2006|p=332}}</ref><ref name="IwaniecKowalski84">{{harvnb|Iwaniec|Kowalski|2004|p=84}}</ref>

Another way to state the functional equation is in terms of
:<math>\Lambda(s,\chi) = q ^{s/2} \pi^{-(s+\delta)/2} \operatorname{\Gamma}\left(\frac{s+\delta}{2}\right) L(s,\chi).</math>
The functional equation can be expressed as:<ref name="MontgomeryVaughan333" /><ref name="IwaniecKowalski84" />
:<math>\Lambda(s,\chi) = W(\chi) \Lambda(1-s,\overline{\chi}).</math>

The functional equation implies that <math>L(s,\chi)</math> (and <math>\Lambda(s,\chi)</math>) are [[entire function|entire functions]] of ''s''. (Again, this assumes that ''χ'' is primitive character modulo ''q'' with ''q'' > 1. If ''q'' = 1, then <math>L(s,\chi) = \zeta(s)</math> has a pole at ''s'' = 1.)<ref name="MontgomeryVaughan333">{{harvnb|Montgomery|Vaughan|2006|p=333}}</ref><ref name="IwaniecKowalski84" />

For generalizations, see: [[Functional equation (L-function)]].