Formula:DLMF:25.15:E4

<< Formula:DLMF:25.15:E3

formula in Dirichlet L-functions

Formula:DLMF:25.15:E5 >>

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=L\left(s,\chi_{0}\right)\prod_{p\mathbin{|}k}\left(1-\frac{\chi_{0}(p)}{p^{s}}\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\chi_{0}}}}$ is a primitive character (mod ${\displaystyle{\displaystyle{\displaystyle d}}}$) for some positive divisor ${\displaystyle{\displaystyle{\displaystyle d}}}$ of ${\displaystyle{\displaystyle{\displaystyle k}}}$ &
hold for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ if ${\displaystyle{\displaystyle{\displaystyle\chi\neq\chi_{1}}}}$, and for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ (${\displaystyle{\displaystyle{\displaystyle\neq 1}}}$) if ${\displaystyle{\displaystyle{\displaystyle\chi=\chi_{1}}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle L}}}$ : Dirichlet ${\displaystyle{\displaystyle{\displaystyle L}}}$-function : http://dlmf.nist.gov/25.15#E1
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod
${\displaystyle{\displaystyle{\displaystyle\nabla\cdot}}}$ : divides : http://dlmf.nist.gov/24.1

Bibliography

Equation (4), Section 25.15 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:DLMF:25.15:E3

formula in Dirichlet L-functions

Formula:DLMF:25.15:E5 >>