Dirichlet L-functions

E259755

L-function complex analytic function object of analytic number theory

Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.

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All labels observed (5)

Label	Occurrences
Dirichlet L-functions canonical	9
Dirichlet L-function	3
Dirichlet L-function Euler products	1
Dirichlet L-function modulo q	1
Dirichlet L-functions modulo q	1

Statements (48)

Predicate	Object
instanceOf	L-function ⓘ complex analytic function ⓘ object of analytic number theory ⓘ
basedOn	Dirichlet characters ⓘ
centralIn	Dirichlet's theorem on arithmetic progressions ⓘ surface form: Dirichlet’s theorem on arithmetic progressions
conjecture	generalized Riemann hypothesis ⓘ surface form: Generalized Riemann Hypothesis for Dirichlet L-functions
convergesAbsolutelyOn	half-plane Re(s)>1 ⓘ
dependsOn	modulus q of Dirichlet character ⓘ primitive or imprimitive nature of character χ ⓘ
domain	complex plane ⓘ
encodes	arithmetic information about residue classes modulo q ⓘ
extendsTo	meromorphic function on C ⓘ
generalizes	Riemann zeta function ⓘ
GRHStatement	all nontrivial zeros lie on line Re(s)=1/2 ⓘ
hasCoefficient	Dirichlet character values χ(n) ⓘ
hasComponent	completed L-function Λ(s,χ) ⓘ
hasEulerProduct	L(s,χ)=∏_{p}(1−χ(p)p^{-s})^{-1} for Re(s)>1 ⓘ
hasGeneralization	L-functions attached to Grössencharacters ⓘ
hasSeriesDefinition	L(s,χ)=∑_{n=1}^{∞} χ(n)n^{-s} for Re(s)>1 ⓘ
hasSpecialCase	Dirichlet beta function as L(s,χ) for nontrivial character mod 4 ⓘ Riemann zeta function ⓘ surface form: Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1
hasSymmetry	functional equation symmetric about Re(s)=1/2 ⓘ
hasTool	explicit formula relating zeros and primes in arithmetic progressions ⓘ
hasZeroType	nontrivial zeros ⓘ trivial zeros ⓘ
implies	infinitely many primes in each admissible arithmetic progression ⓘ
involves	Gamma factors in functional equation ⓘ conductor of Dirichlet character ⓘ parity of Dirichlet character ⓘ
multiplicativityProperty	Dirichlet characters are completely multiplicative ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet ⓘ
nontrivialZerosRegion	critical strip 0<Re(s)<1 ⓘ
nonvanishingProperty	L(1,χ)≠0 for nontrivial Dirichlet characters χ ⓘ
poleProperty	L(s,χ) is entire for nonprincipal characters ⓘ
relatedTo	Artin L-functions ⓘ L-functions ⓘ surface form: Hecke L-functions automorphic L-functions ⓘ
satisfies	analytic continuation except possible simple pole at s=1 for principal characters ⓘ functional equation relating L(s,χ) and L(1−s,χ̄) ⓘ
studiedIn	analytic number theory ⓘ
trivialZerosLocation	negative integers depending on parity of χ ⓘ
usedIn	Chebotarev density theorem ⓘ analytic class number formula for imaginary quadratic fields ⓘ class number formulas for quadratic fields ⓘ distribution of primes in residue classes ⓘ study of primes in arithmetic progressions ⓘ
usedToDefine	Dirichlet density of sets of primes ⓘ
variable	complex variable s ⓘ

Referenced by (15)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann hypothesis → relatedTo → Dirichlet L-functions ⓘ

Riemann zeta function → generalization → Dirichlet L-functions ⓘ

Euler product formula for the Riemann zeta function → relatesTo → Dirichlet L-functions ⓘ

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet L-functions ⓘ

Multiplicative Number Theory → relatedTo → Dirichlet L-functions ⓘ

Legendre symbol → usedIn → Dirichlet L-functions ⓘ

Deuring–Heilbronn phenomenon → involves → Dirichlet L-functions ⓘ

this entity surface form: Dirichlet L-function

Deuring–Heilbronn phenomenon → appliesTo → Dirichlet L-functions ⓘ

this entity surface form: Dirichlet L-function modulo q

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function → relatedConcept → Dirichlet L-functions ⓘ

Selberg class → contains → Dirichlet L-functions ⓘ

generalized Riemann hypothesis → statementAbout → Dirichlet L-functions ⓘ

generalized Riemann hypothesis → appliesTo → Dirichlet L-functions ⓘ

this entity surface form: Dirichlet L-functions modulo q

Dirichlet series → specialCase → Dirichlet L-functions ⓘ

this entity surface form: Dirichlet L-function

Euler products for automorphic L-functions → generalizes → Dirichlet L-functions ⓘ

this entity surface form: Dirichlet L-function Euler products

L-functions → hasSpecialCase → Dirichlet L-functions ⓘ

subject surface form: L-function

this entity surface form: Dirichlet L-function