Definition:Completed Dirichlet L-Function

This article needs to be linked to other articles. In particular: modulus, principal character You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Definition

Let $\chi$ be a primitive Dirichlet character to the modulus $q \ge 1$.

Let $\kappa = \dfrac 1 2 \paren {1 - \map \chi {-1} }$.

Let $\delta = 1$ if $\chi$ is the principal character, and $0$ otherwise.

The completed Dirichlet $L$-function for $\chi$ is defined to be

$\map \Lambda {s, \chi} = \dfrac {1 + \kappa} 2 \paren {s \paren {1 - s} }^\delta \map \Gamma {\dfrac {s + \kappa} 2} \map L {s, \chi}$

where $\map L {s, \chi}$ is the Dirichlet $L$-function for $\chi$.