An exceptional zero is the possible real simple zero omitted from the otherwise uniform Classical zero-free region for Dirichlet L-functions. It belongs to a real nonprincipal primitive Dirichlet character and satisfies .
Uniformly for and ,where the second term occurs only when a real character has an exceptional zero .
If some satisfiesuniformly for sufficiently large , then every exceptional zero modulo satisfiesChoose and in the exceptional-zero asymptotic, with large in terms of .
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The Siegel zero is a concept in number theory, particularly in the field of analytic number theory. It refers to a hypothetical zero of a certain class of Dirichlet L-functions, specifically those associated with non-principal characters of a Dirichlet character modulo \( q \). The Siegel zero is named after Carl Ludwig Siegel, who studied these functions.