Self-approximation of Dirichlet L-functions

Let d be a real number, let s be in a fixed compact set of the strip 1/2<σ<1, and let L(s,χ) be the Dirichlet L-function. The hypothesis is that for any real number d there exist ‘many’ real numbers τ such that the shifts L(s+iτ,χ) and L(s+idτ,χ) are ‘near’ each other. If d is an algebraic irrational number then this was obtained by T. Nakamura. Ł. Pańkowski solved the case then d is a transcendental number. We prove the case then d≠0 is a rational number. If d=0 then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet L-function. We also consider a more general version of the above problem.