Formula:DLMF:25.15:E5

<< Formula:DLMF:25.15:E4

formula in Dirichlet L-functions

Formula:DLMF:25.15:E7 >>

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle G(\chi) = \sum_{r=1}^k \chi(r) \expe^{2 \cpi \iunit r/k}}}}

Constraint(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \chi}} is a primitive character (mod Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k}} ) &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \overline{\chi}}} is the complex conjugate of Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \chi}}

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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle L}}  : Dirichlet Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \DirichletL}} -function : http://dlmf.nist.gov/25.15#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Gamma}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pi}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{e}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{i}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum

Bibliography

Equation (5), Section 25.15 of DLMF.

URL links

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

<< Formula:DLMF:25.15:E4

formula in Dirichlet L-functions

Formula:DLMF:25.15:E7 >>