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DirichletL[k,j,s]

gives the Dirichlet L-function for the Dirichlet character with modulus k and index j.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Neat Examples  
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DirichletL

DirichletL[k,j,s]

gives the Dirichlet L-function for the Dirichlet character with modulus k and index j.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , , where is a Dirichlet character.
  • The possible Dirichlet characters modulo k are specified by an index j, and given by DirichletCharacter[k,j,n].
  • DirichletL[k,j,s] can be evaluated to arbitrary numerical precision for integer k and j, and arbitrary complex s.
  • DirichletL automatically threads over lists.
  • DirichletL can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (2)

Compute an exact value:

Wolfram Language code: DirichletL[1, 1, 2]

A plot along the line :

Wolfram Language code: Plot[Re@DirichletL[2, 1, 1 + I ω], {ω, 1, 100}]

Scope  (5)

Evaluate for complex arguments:

Wolfram Language code: DirichletL[2, 1, 1. + I]

Evaluate to high precision:

Wolfram Language code: N[DirichletL[3, 2, 1 / 2], 50]

The precision of the output tracks the precision of the input:

Wolfram Language code: DirichletL[3, 2, 1.50000000000000000000000]

Simple exact values are generated automatically:

Wolfram Language code: Table[DirichletL[1, 1, s], {s, -6, 6}]

DirichletL threads element-wise over lists and matrices:

Wolfram Language code: DirichletL[1, 1, {1, 2, 3, 4}]

DirichletL can be used with Interval and CenteredInterval objects:

Wolfram Language code: DirichletL[5, 1, Interval[{1.23, 1.24}]]
Wolfram Language code: DirichletL[3, 2, CenteredInterval[3 / 2, 1 / 100]]

Applications  (4)

Plot the real part of a DirichletL function on the critical line:

Wolfram Language code: Plot[Re[DirichletL[8, 3, 1 / 2 + I t]], {t, 0, 50}]

Plot the real part across the critical strip:

Wolfram Language code: Plot3D[Re[DirichletL[8, 3, s + I t]], {s, 0, 1}, {t, 0, 50}, BoxRatios -> {1, 5, 1.5}, Mesh -> None, PlotPoints -> {15, 50}]

Find a zero of a DirichletL function:

Wolfram Language code: FindRoot[DirichletL[8, 3, t] == 0, {t, 19I}]

Plot real and imaginary parts in the vicinity of nearby zeros:

Wolfram Language code: Plot[{Re[DirichletL[8, 3, 1 / 2 + I t]], Im[DirichletL[8, 3, 1 / 2 + I t]]}, {t, 40, 46}]

Neat Examples  (1)

The real part of a Dirichlet L-function:

Wolfram Language code: Plot3D[Re@DirichletL[10, 1, u + I v], {u, -5, 5}, {v, -10, 10}, Mesh -> None, PlotStyle -> Directive[Opacity[0.7], Orange, Specularity[10]], BoxRatios -> {1, 1, 2}]
Wolfram Research (2008), DirichletL, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletL.html (updated 2023).

Text

Wolfram Research (2008), DirichletL, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletL.html (updated 2023).

CMS

Wolfram Language. 2008. "DirichletL." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/DirichletL.html.

APA

Wolfram Language. (2008). DirichletL. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DirichletL.html

BibTeX

@misc{reference.wolfram_2026_dirichletl, author="Wolfram Research", title="{DirichletL}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletL.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dirichletl, organization={Wolfram Research}, title={DirichletL}, year={2023}, url={https://reference.wolfram.com/language/ref/DirichletL.html}, note=[Accessed: 13-June-2026]}