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In [[mathematics]], a '''Dirichlet ''L''-series''' is a function of the form

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math>

Here χ is a [[Dirichlet character]] and ''s'' a [[complex variable]] with [[real part]] greater than 1. By [[analytic continuation]], this function can be extended to a [[meromorphic function]] on the whole [[complex plane]], and is then called a '''Dirichlet ''L''-function''' and also denoted ''L''(''s'', χ).

These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in {{harv|Dirichlet|1837}} to prove the [[Dirichlet's theorem on arithmetic progressions|theorem on primes in arithmetic progressions]] that also bears his name. In the course of the proof, Dirichlet shows that {{Nowrap|''L''(''s'', χ)}} is non-zero at ''s'' = 1. Moreover, if χ is principal, then the corresponding Dirichlet ''L''-function has a [[simple pole]] at ''s'' = 1.

== Zeros of the Dirichlet L-functions ==

If χ is a primitive character with χ(&minus;1) = 1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative even integers.
If χ is a primitive character with χ(&minus;1) = &minus;1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative odd integers.

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>

Just as the Riemann zeta function is conjectured to obey the [[Riemann hypothesis]], so the Dirichlet ''L''-functions are conjectured to obey the [[generalized Riemann hypothesis]].

== Euler product ==
Since a Dirichlet character χ is [[completely multiplicative]], its ''L''-function can also be written as an [[Euler product]] in the [[half-plane]] of [[absolute convergence]]:
:<math>L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,</math>
where the product is over all [[prime number]]s.<ref>{{harvnb|Apostol|1976|loc=Theorem 11.7}}</ref>

== Functional equation ==

Let us assume that χ is a primitive character to the modulus ''k''. Defining

:<math>\Lambda(s,\chi) = \left(\frac{\pi}{k}\right)^{-(s+a)/2}
\Gamma\left(\frac{s+a}{2}\right) L(s,\chi),</math>

where Γ denotes the [[Gamma function]] and the symbol ''a'' is given by

:<math>a=\begin{cases}0;&\mbox{if }\chi(-1)=1, \\ 1;&\mbox{if }\chi(-1)=-1,\end{cases}</math>

one has the [[functional equation (L-function)|functional equation]]

:<math>\Lambda(1-s,\overline{\chi})=\frac{i^ak^{1/2}}{\tau(\chi)}\Lambda(s,\chi),</math>

where τ(χ) is the [[Gauss sum]]

:<math>\sum_{n=1}^k\chi(n)\exp(2\pi in/k).</math>

Note that |τ(χ)| = ''k''<sup>1/2</sup>.

== Relation to the Hurwitz zeta-function ==
The Dirichlet ''L''-functions may be written as a linear combination of the [[Hurwitz zeta function|Hurwitz zeta-function]] at rational values. Fixing an integer ''k'' ≥ 1, the Dirichlet ''L''-functions for characters modulo ''k'' are linear combinations, with constant coefficients, of the ζ(''s'',''q'') where ''q'' = ''m''/''k'' and ''m'' = 1, 2, ..., ''k''. This means that the Hurwitz zeta-function for rational ''q'' has analytic properties that are closely related to the Dirichlet ''L''-functions. Specifically, let &chi; be a character modulo ''k''.  Then we can write its Dirichlet ''L''-function as

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac {\chi(n)}{n^s}
= \frac {1}{k^s} \sum_{m=1}^k \chi(m)\; \zeta \left(s,\frac{m}{k}\right).</math>

==See also==

*[[Generalized Riemann hypothesis]]
*[[L-function]]
*[[Modularity theorem]]
*[[Artin conjecture (L-functions)|Artin conjecture]]
*[[Special values of L-functions]]

==Notes==
{{reflist}}

== References ==
* {{Apostol IANT}}
*{{dlmf|id=25.15|first=T. M.|last=Apostol}}
* {{cite book|author=H. Davenport
| title=Multiplicative Number Theory
|publisher=Springer
|year=2000
|isbn=0-387-95097-4}}
* {{Cite journal
| last=Dirichlet
| first=P. G. L.
| author-link=Peter Gustav Lejeune Dirichlet
| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
| journal=Abhand. Ak. Wiss. Berlin
| volume=48
| year=1837
| ref=harv
| postscript=.
}}
* {{springer|title=Dirichlet-L-function|id=p/d032890}}

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[[Category:Zeta and L-functions]]