  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 $s = 1/2.$
