Dirichlet L-functions

Definitions and Basic Properties

$\displaystyle {\displaystyle \DirichletL@{s}{\chi} = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$ &
$\displaystyle {\displaystyle \chi(n)}$ is a Dirichlet character~$\displaystyle {\displaystyle \pmod{k}}$

$\displaystyle {\displaystyle \DirichletL@{s}{\chi} = \prod_p \left( 1 - \frac{\chi(p)}{p^s} \right)^{-1} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$

$\displaystyle {\displaystyle \DirichletL@{s}{\chi} = k^{-s} \sum_{r=1}^{k-1} \chi(r) \HurwitzZeta@{s}{\frac{r}{k}} }$

Constraint(s): hold for all $\displaystyle {\displaystyle s}$ if $\displaystyle {\displaystyle \chi \neq \chi_1}$ , and for all $\displaystyle {\displaystyle s}$ ($\displaystyle {\displaystyle \neq 1}$ ) if $\displaystyle {\displaystyle \chi = \chi_1}$

$\displaystyle {\displaystyle \DirichletL@{s}{\chi} = \DirichletL@{s}{\chi_0} \prod_{p \divides k} \left( 1 - \frac{\chi_0(p)}{p^s} \right) }$

Constraint(s): $\displaystyle {\displaystyle \chi_0}$ is a primitive character (mod $\displaystyle {\displaystyle d}$ ) for some positive divisor $\displaystyle {\displaystyle d}$ of $\displaystyle {\displaystyle k}$ &
hold for all $\displaystyle {\displaystyle s}$ if $\displaystyle {\displaystyle \chi \neq \chi_1}$ , and for all $\displaystyle {\displaystyle s}$ ($\displaystyle {\displaystyle \neq 1}$ ) if $\displaystyle {\displaystyle \chi=\chi_1}$

$\displaystyle {\displaystyle \DirichletL@{1-s}{\chi} = \frac{k^{s-1} \EulerGamma@{s}}{(2 \cpi)^s} \* {\left( \expe^{-\cpi \iunit s/2} + \chi(-1) \expe^{\cpi \iunit s/2} \right)} \* G(\chi) \DirichletL@{s}{\overline{\chi}} }$

Substitution(s): $\displaystyle {\displaystyle {\displaystyle G(\chi) = \sum_{r=1}^k \chi(r) \expe^{2 \cpi \iunit r/k}}}$

Constraint(s): $\displaystyle {\displaystyle \chi}$ is a primitive character (mod $\displaystyle {\displaystyle k}$ ) &
$\displaystyle {\displaystyle \overline{\chi}}$ is the complex conjugate of $\displaystyle {\displaystyle \chi}$

Zeros

$\displaystyle {\displaystyle \DirichletL@{-2n}{\chi} \hiderel{=} 0 \text{\quad if\quad} \chi(-1) \hiderel{=} 1 }$

Constraint(s): $\displaystyle {\displaystyle n = 0,1,2,\dots}$

$\displaystyle {\displaystyle \DirichletL@{-2n-1}{\chi} \hiderel{=} 0 \text{\quad if\quad} \chi(-1) \hiderel{=} -1 }$

Constraint(s): $\displaystyle {\displaystyle n = 0,1,2,\dots}$

$\displaystyle {\displaystyle \DirichletL@{1}{\chi} \hiderel{\neq} 0 \text{\quad if\quad} \chi \hiderel{\neq} -1 }$

Constraint(s): $\displaystyle {\displaystyle \chi_1}$ is the principal character~$\displaystyle {\displaystyle \pmod{k}}$

$\displaystyle {\displaystyle \DirichletL@{0}{\chi} = \begin{cases} \displaystyle -\frac{1}{k} \sum_{r=1}^k r \chi(r), & \chi \neq \chi_1, \\ 0, & \chi = \chi_1. \end{cases} }$