Editing Dirichlet L-function (section)

==Primitive characters==

Results about ''L''-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications.<ref>{{harvnb|Davenport|2000|loc=chapter 5}}</ref> This is because of the relationship between a imprimitive character <math>\chi</math> and the primitive character <math>\chi^\star</math> which induces it:<ref>{{harvnb|Davenport|2000|loc=chapter 5, equation (2)}}</ref>
:<math>
  \chi(n) =
    \begin{cases}
      \chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\
      0, & \mathrm{if} \gcd(n,q) \ne 1
    \end{cases}
</math>
(Here, ''q'' is the modulus of ''χ''.) An application of the Euler product gives a simple relationship between the corresponding ''L''-functions:<ref>{{harvnb|Davenport|2000|loc=chapter 5, equation (3)}}</ref><ref>{{harvnb|Montgomery|Vaughan|2006|p=282}}</ref>
:<math>
  L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \frac{\chi^\star(p)}{p^s} \right)
</math>
(This formula holds for all ''s'', by analytic continuation, even though the Euler product is only valid when Re(''s'') > 1.) The formula shows that the ''L''-function of ''χ'' is equal to the ''L''-function of the primitive character which induces ''χ'', multiplied by only a finite number of factors.<ref>{{harvnb|Apostol|1976|p=262}}</ref>

As a special case, the ''L''-function of the principal character <math>\chi_0</math> modulo ''q'' can be expressed in terms of the [[Riemann zeta function]]:<ref>{{harvnb|Ireland|Rosen|1990|loc=chapter 16, section 4}}</ref><ref>{{harvnb|Montgomery|Vaughan|2006|p=121}}</ref>
:<math>
  L(s,\chi_0) = \zeta(s) \prod_{p \,|\, q}(1 - p^{-s})
</math>