Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to a variant of the Weil distribution which occurs in André Weil's explicit formula. Hence this representation encodes information about the distribution of the prime ideals of K and is a spectral interpretation for the poles and zeros of the L-function of K. Our construction is motivated by a similar spectral interpretation by Alain Connes.
Ralf Meyer. "On a representation of the idele class group related to primes and zeros of L-functions." Duke Math. J. 127 (3) 519 - 595, 15 April 2005. https://doi.org/10.1215/S0012-7094-04-12734-4