Dedekind zeta functions

E262117

Dirichlet series L-function number-theoretic function

Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.

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All labels observed (3)

Label	Occurrences
Dedekind zeta functions canonical	2
Dedekind zeta function	1
Dedekind zeta function for number fields	1

Statements (49)

Predicate	Object
instanceOf	Dirichlet series ⓘ L-function ⓘ number-theoretic function ⓘ
appearsIn	analytic class number formula ⓘ proofs of Dirichlet unit theorem ⓘ proofs of finiteness of class number ⓘ
associatedWith	ring of integers of a number field ⓘ
conjecturallySatisfies	generalized Riemann hypothesis for number fields ⓘ
definedBy	Dirichlet series over nonzero ideals of the ring of integers of a number field ⓘ
definedOn	algebraic number field ⓘ
dependsOn	degree of the number field ⓘ discriminant of the number field ⓘ signature of the number field ⓘ
domain	complex plane ⓘ
encodes	arithmetic properties of number fields ⓘ class numbers ⓘ discriminant of a number field ⓘ distribution of prime ideals ⓘ unit group information ⓘ
extendedBy	analytic continuation to all complex s except a pole ⓘ
fieldOfStudy	algebraic number theory ⓘ analytic number theory ⓘ
generalizes	Riemann zeta function ⓘ
hasConvergenceRegion	Re(s) > 1 ⓘ
hasEulerProduct	product over prime ideals ⓘ
hasInvariant	residue at s = 1 ⓘ
hasPoleAt	s = 1 ⓘ
hasProperty	analytic continuation ⓘ functional equation ⓘ meromorphic function of s ⓘ
hasVariable	complex variable s ⓘ
hasZeroType	nontrivial zeros ⓘ trivial zeros ⓘ
namedAfter	Richard Dedekind ⓘ
orderOfPoleAt	1 at s = 1 ⓘ
relatedTo	Artin L-functions ⓘ Chebotarev density theorem ⓘ L-functions ⓘ surface form: Hecke L-functions class number formula ⓘ prime ideal theorem ⓘ
satisfies	Euler product over prime ideals of the ring of integers ⓘ
specialCase	Riemann zeta function for the rational field ⓘ
usedIn	algebraic number theory ⓘ class field theory ⓘ distribution of splitting of primes in extensions ⓘ study of discriminants ⓘ
usedToDefine	class number of a number field ⓘ regulator of a number field ⓘ residue at s = 1 expressing class number and regulator ⓘ

Referenced by (4)

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

Riemann zeta function → generalization → Dedekind zeta functions ⓘ

Hasse–Weil zeta function → specialCase → Dedekind zeta functions ⓘ

this entity surface form: Dedekind zeta function for number fields

Selberg class → generalizes → Dedekind zeta functions ⓘ

L-functions → hasSpecialCase → Dedekind zeta functions ⓘ

subject surface form: L-function

this entity surface form: Dedekind zeta function