We consider the Schrödinger operator −∆ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (−∆ + V)ψ = 0, namely the kernel of (−∆ + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n ≥ 5 and in dimension three we generalize a result proved by Beceanu.