An L-function is an arithmetic function given initially by a Dirichlet series and often an Euler product, with analytic continuation and a functional equation in important cases. Dirichlet L-functions and L-functions of cusp forms are basic examples; p-adic L-functions interpolate suitably modified special values.
A p-adic L-function is a -adic analytic function or measure interpolating suitably adjusted special values of an L-function. The Kubota-Leopoldt p-adic L-function is the basic example attached to a Dirichlet character.
For an even Dirichlet character , the Kubota-Leopoldt function interpolates generalized Bernoulli values with the relevant Euler factor and Teichmüller character adjustment. If , it can be encoded by an integral power series. For , the -ramified convention is . The trivial character requires separate treatment of the pole.
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L-functions are a broad class of complex functions that arise in number theory and are connected to various areas of mathematics, including algebraic geometry, representation theory, and mathematical physics. The concept of an L-function is primarily associated with the study of prime numbers and solutions to polynomial equations, and they encapsulate deep properties of arithmetic objects.