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An (analytic) L-function is a Dirichlet series that has an Euler product and satisfies a certain type of functional equation.

It is expected that all L-functions satisfy the Riemann Hypothesis, that all of the zeros in the critical strip are on the critical line. Selberg has defined a class $\mathcal S$ of Dirichlet series that satisfy the Selberg axioms. It is conjectured (but far from proven) that $\mathcal S$ is precisely the set of all L-functions. Selberg's axioms have not been verified for all of the L-functions in this database but are known to hold for many of them.

It is also conjectured that a precise form of the functional equation holds for every element of $\mathcal S$. Under this assumption the functional equation is determined by a quadruple known as the Selberg data, consisting of the degree, conductor, spectral parameters, and sign.

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• Last edited by Andrew Sutherland on 2019-05-19 08:43:41
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