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Let K be a global field. Using natural spaces of functions on the adele ringand the idele class group of K, we construct a virtual representation of theidele class group of K whose character is equal to the Weil distribution thatoccurs in Andre Weil's Explicit Formula. Hence this representation encodesinformation about the distribution of the prime ideals of K, and it is aspectral interpretation for the poles and zeros of the L-function. It isclosely related to the spectral interpretation of Alain Connes

On a representation of the idele class group related to primes and zeros of L-functions