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Weil indices are attached individually to local Well representations of local symplectic groups and, collectively (and in a product formula), to the adélic Weil Θ funtional. Therefore Weil indices can be attached to classical ζ functions. Root numbers, or local constants, are attached to the decomposition of L and functions into local factors. Since there is a classical integral-transform connection between ζ and ζ there must be an according conection between Weil indices and local constants. We prove that this is so for the prototypical case of the Dedekind ζ funtion, and, indeed, that this relationship is given in terms of an equation involving an integral of dimension nplus1 where n is the degree of the underlying (totally real) algebraic number field
Berg, M. “On Local Objects Attached to Theta- and Zeta-Functions," Journal of Integral Transforms and Special functions, 10(1), 2000, 13-24.