Artin L-functions are a family of complex functions associated with Galois representations and algebraic number fields, playing a crucial role in understanding the distribution of prime ideals and the behavior of number-theoretic properties. They extend the idea of Dirichlet L-functions to more general contexts, encapsulating important arithmetic information about field extensions and their Galois groups. These functions are central to many deep conjectures in number theory, such as the Langlands program and the proof of the class number formula.
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