Super-positivity of a family of L-functions in the level aspect

Abstract

An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point $s>1/2$ are positive. In this paper we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight $2$ and large prime level $q$ have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $\Re (s)>0$ except possibly at $s=1/2.$

28 pages. For Don Zagier on his 65th birthday. To appear in Research in the Mathematical Sciences. Comments are welcome

References in corpus (1)

• Super-positivity of a family of L-functions