Nonzero values of Dirichlet $L$-functions in vertical arithmetic progressions

Let $L(s,\chi )$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\mathrm{\Re }(s)=1\mathrm{/}2$, we show that $\gg T\log T$ of them are not zeros of $L(s,\chi )$. This result provides some theoretical evidence towards the conjecture that all ordinates of zeros of Dirichlet $L$-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of $L(s,\chi )$.