Zeroes of L-functions are conjectured to have spectral interpretation and the symmetry of a family of L-function linked to the distribution of zeros hence assumed to come from some suitable actions. In the following letter, Sarnak defines a symplectic pairing on a space of distributions (due to Connes) which annihilate some theta functions corresponding to a quadratic Dirichlet characters.
http://publications.ias.edu/sites/default/files/BombieriLtr2002.pdf
A Neighbourhood Of Infinity 