Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated intensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by -A which is only logarithmically stronger than the usual condition for the unitary group generated by iA. This condition is sharp when the observation operator is bounded. The proof combines the so-called ''control transmutation method'' and a new version of the ''direct Lebeau-Robbiano strategy''. The improvement of this strategy also yields interior null-controllability of new logarithmic anomalous diffusions.