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DirichletL ​

The Dirichlet L-function of the -th character mod — the family that contains , and , and the setting of Dirichlet's theorem on primes in arithmetic progressions. Provided by `@enumeratio/analytic`.

WikipediaDirichlet L-functionMathWorldDirichletL-SeriesWikidataQ2264419nLabDirichlet L-functionDLMF25.15.E1FungrimDirichletLFungrim entry3b8c97a07d28a9337bc9d117d83109ff8254Wolfram LanguageDirichletL
DirichletL(k, j, s)the L-function of DirichletCharacter .

Domain: Special functions

Details
  • , and for the principal character mod the Euler factors at the primes dividing drop out: — so it inherits 's pole at . Every non-principal is entire.
  • , the DirichletBeta function; is not an L-function of this family (the alternating sign is not a character mod 2) but is . See DirichletEta.
  • for every non-principal is the analytic heart of Dirichlet's theorem: each residue class coprime to contains infinitely many primes.
  • Values at nonpositive integers are exact: , an algebraic number in the character's roots of unity (generalized Bernoulli numbers). See BernoulliB.
  • Numerically, on the HurwitzZeta kernel, for complex ; near the Hurwitz poles cancel against each other, so there the Laurent expansion in the generalized StieltjesGamma constants is summed instead.

Examples

Implementation

referenceenginenotatio

The Hurwitz decomposition — the definition the kernel evaluates, away from s = 1.

nativeenginetypescriptpackages/analytic/src/dirichlet-l.ts
mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

DirichletL[k, j, s]; mpmath has dirichlet(s, chi) but takes the character as a period, with no indexing of its own.

See also: DirichletCharacter, DirichletBeta, DirichletEta, HurwitzZeta, Zeta