# Booker’s Extension of the Selberg Class

Briefly, the Selberg class is a set of functions $F:\C\to\C$ such that $f(s)$ can be written as a Dirichlet series for $\Re(s) > 1$ and that satisfies a form of analytic continuation, a functional equation, a Ramanujan hypothesis bound on coefficients of the Dirichlet series, and an Euler product formula.

Andrew Booker in [1] has extended the Selberg class in a different way, in the notion of an $L$-datum. In this post, we’ll state Booker’s definition of an $L$-datum, state his converse theorem, and explain his corollary that the completed $L$-function of a unitary cuspidal automorphic representation of $\GL_3(\A_\Q)$ has infinitely many zeroes of odd order.
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