DirichletL

The Dirichlet L-function $L(s, \chi) = \sum_{n\ge1} \chi(n)\,n^{-s}$ of the $j$-th character mod $k$ — the family that contains $\zeta$, $\eta$ and $\beta$, and the setting of Dirichlet's theorem on primes in arithmetic progressions.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDirichlet L-function↗MathWorldDirichletL-Series↗WikidataQ2264419↗nLabDirichlet L-function↗DLMF25.15.E1↗FungrimDirichletL↗Wolfram LanguageDirichletL!↗

DirichletL(k, j, s)the L-function of DirichletCharacter $\chi_j \bmod k$.

Typed signatures

Type	Package	Applies
(integer, integer, number) -> number	analyticreplaces compute-engine	always
(integer, integer, number) -> number	compute-engine	always

Domain: Special functions

Details

• $L(s, \chi_1 \bmod 1) = \zeta(s)$, and for the principal character mod $k$ the Euler factors at the primes dividing $k$ drop out: $L(s, \chi_1) = \zeta(s)\prod_{p \mid k}(1 - p^{-s})$ — so it inherits $\zeta$'s pole at $s = 1$. Every non-principal $L$ is entire.
• $L(s, \chi_2 \bmod 4) = \beta(s)$, the DirichletBeta function; $\eta(s)$ is not an L-function of this family (the alternating sign is not a character mod 2) but is $(1 - 2^{1-s})\zeta(s)$. See DirichletEta.
• $L(1, \chi) \neq 0$ for every non-principal $\chi$ is the analytic heart of Dirichlet's theorem: each residue class coprime to $k$ contains infinitely many primes.
• Values at nonpositive integers are exact: $L(-n, \chi) = -k^n \sum_{r=1}^{k} \chi(r)\,B_{n+1}(r/k)/(n+1)$, an algebraic number in the character's roots of unity (generalized Bernoulli numbers). See BernoulliB.
• Numerically, $L(s, \chi) = k^{-s}\sum_{r=1}^{k}\chi(r)\,\zeta(s, r/k)$ on the HurwitzZeta kernel, for complex $s$; near $s = 1$ the Hurwitz poles cancel against each other, so there the Laurent expansion in the generalized StieltjesGamma constants is summed instead.

Plot

Examples

7 more kept as data, checked against the oracles

Basic #

Modulus 1: the Riemann zeta function itself

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In$\mathrm{DirichletL}(1, 1, s)$Out$ζ(s)$

$L(1, \chi_2 \bmod 4) = \beta(1) = \pi/4$ — Leibniz's series

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In$\mathrm{DirichletL}(4, 2, 1)$Out$\frac{π}{4}$

$L(2, \chi_1 \bmod 1) = \zeta(2) = \pi^2/6$

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In$\mathrm{DirichletL}(1, 1, 2)$Out$\frac{π^2}{6}$

Numeric for a floating-point argument

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In$\mathrm{DirichletL}(3, 2, 1.5)$Out$0.703\,968\,244\,868\,732\,9$

Scope #

Complex $s$: $L(s, \chi_1 \bmod 2) = (1 - 2^{-s})\zeta(s)$ at $s = 1 + i$

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In$N(\mathrm{DirichletL}(2, 1, 1+i))$Out$0.654\,358\,917\,407\,233\,1-0.384\,376\,350\,214\,799\,6i$

Listable in $s$: threads over a list

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In$\mathrm{DirichletL}(1, 1, \bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack\tilde∞, \frac{π^2}{6}, ζ(3), \frac{π^4}{90}\bigr\rbrack$

Interval arithmetic in $s$ (decreasing here)

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In$\mathrm{DirichletL}(5, 1, \mathrm{Interval}(1.23, 1.24))$Out$\mathrm{Interval}(4.113\,958\,567\,039\,058, 4.258\,988\,950\,108\,49)$

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In$N(\mathrm{DirichletL}(3, 2, \frac{1}{2}), 50)$Out$0.480\,867\,557\,696\,828\,626\,181\,220\,063\,235\,589\,877\,776\,829\,730\,832\,37$

Properties #

A principal character drops the Euler factors at $2$ and $3$

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In$\mathrm{DirichletL}(12, 1, s)$Out$(1-2^{-s})(1-3^{-s})ζ(s)$

…and keeps $\zeta$'s pole; a non-principal $L$ is entire

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In$\mathrm{DirichletL}(12, 1, 1)$Out$\tilde∞$

$L(0, \chi) = \tfrac35 + \tfrac15 i$ — exact, from the generalized Bernoulli numbers

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In$\mathrm{DirichletL}(5, 2, 0)$Out$\frac{3}{5}+\frac{1}{5}i$

$L(-3, \chi_2 \bmod 8) = 11$

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In$\mathrm{DirichletL}(8, 2, -3)$Out$11$

The principal character mod 2 removes the Euler factor at 2

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In$\mathrm{DirichletL}(2, 1, s)$Out$(1-2^{-s})ζ(s)$

$L(2, \chi_2 \bmod 4) = \beta(2) = G$

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In$\mathrm{DirichletL}(4, 2, 2)$Out$\operatorname{G}$

$L(0, \chi_{-3}) = \tfrac13$ — the class number formula's $2h/w$ for $\mathbb{Q}(\sqrt{-3})$

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In$\mathrm{DirichletL}(3, 2, 0)$Out$\frac{1}{3}$

$L(1, \chi_{-3}) = \pi/(3\sqrt3)$: $s = 1$ for a real (quadratic) odd character has a closed form, from the Fourier expansion of $\sum \chi(n)/n$

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In$\mathrm{DirichletL}(3, 2, 1)$Out$\frac{\sqrt{3}π}{9}$

The principal character mod 2 gives $\frac{\pi^2}{8}$ at $s=2$

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In$\mathrm{DirichletL}(2, 1, 2)$Out$\frac{π^2}{8}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Odd argument gives a multiple of $\zeta(3)$: $(1-2^{-3})\zeta(3)$

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In$\mathrm{DirichletL}(2, 1, 3)$Out$\frac{7ζ(3)}{8}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Even argument gives a rational multiple of $\pi^4$

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In$\mathrm{DirichletL}(2, 1, 4)$Out$\frac{π^4}{96}$

Known value · Wolfram MathematicalFunctionData, DirichletL

For modulus 3, $s=2$ gives a multiple of $\pi^2$

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In$\mathrm{DirichletL}(3, 1, 2)$Out$\frac{4π^2}{27}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Odd argument gives $\frac{26}{27}\zeta(3)$

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In$\mathrm{DirichletL}(3, 1, 3)$Out$\frac{26ζ(3)}{27}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Modulus 3 at $s=4$ is rational times $\pi^4$

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In$\mathrm{DirichletL}(3, 1, 4)$Out$\frac{8π^4}{729}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Modulus 3 at $s=5$ is $\frac{242}{243}\zeta(5)$

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In$\mathrm{DirichletL}(3, 1, 5)$Out$\frac{242ζ(5)}{243}$

Known value · Wolfram MathematicalFunctionData, DirichletL

For modulus 5, $s=2$ gives $\frac{4\pi^2}{25}$

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In$\mathrm{DirichletL}(5, 1, 2)$Out$\frac{4π^2}{25}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Modulus 5 at $s=3$ is $\frac{124}{125}\zeta(3)$

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In$\mathrm{DirichletL}(5, 1, 3)$Out$\frac{124ζ(3)}{125}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Modulus 6 removes both Euler factors for 2 and 3

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In$\mathrm{DirichletL}(6, 1, 2)$Out$\frac{π^2}{9}$

Known value · Wolfram MathematicalFunctionData, DirichletL

Possible issues #

No closed form at positive integer $s$ in general, so this stays symbolic; N() gives $0.95872 + 0.14557i$

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In$\mathrm{DirichletL}(5, 2, 2)$Out$\mathrm{DirichletL}(5, 2, 2)$

About

Examples	34
Checked against	Wolfram Language 32 agree2 differ
Crosswalk	WikipediaMathWorldWikidatanLabDLMFFungrimWolfram Language

Implementation

referenceenginenotatio

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

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/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