Multiplicity results for the functional equation of the Dirichlet $L$-functions: case $p=2$

Given an integer q and a primitive character χ modulo q, the functionalequation of the Dirichlet L-function L(s, χ) is determined by the signature of χ, i.e. byχ(−1) (the parity) and τ(χ) (the Gauss sum). In this paper we prove several resultsabout the cardinalities of the sets T(χ) := {ψ : τ(ψ) = τ(χ)} and W(χ) := {ψ : τ(ψ) =τ(χ), ψ(−1) = χ(−1)}, mainly an algorithm for their computation and optimal upperand lower bounds for their values, when q is either an odd prime power or a compositenumber of special form. For the same q we compute also the number of distinct Gausssums and of distinct signatures: the latter number deserves a special attention because itcoincides with the number of non-trivial functional equations of degree 1 and conductorq in the Selberg class.