  Let $d\in\mathbb{N}$, and let $\pi$ be a fixed cuspidal automorphic
representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary
central character. We determine the limiting distribution of the family of
values $-\frac{L'}{L}(1+it,\pi\otimes\chi_D)$ as $D$ varies over fundamental
discriminants. Here, $t$ is a fixed real number and $\chi_D$ is the real
character associated with $D$. We establish an upper bound on the discrepancy
in the convergence of this family to its limiting distribution. As an
application of this result, we obtain an upper bound on the small values of
$\left|\frac{L'}{L}(1,\pi\otimes\chi_D)\right|$ when $\pi$ is self-dual.
