We prove a strong form of the trivial zero conjecture at the central point for the -adic -function of a non-critically refined self-dual cohomological cuspidal automorphic representation of over a totally real field, which is Iwahori spherical at places above .
In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the -adic -function of a nearly finite slope family and on improved -adic -functions that we construct using automorphic symbols and overconvergent cohomology.
For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the -adic -function of the family.
