Small values of | L' /L (1, chi) |

In this paper, we investigate the quantity $m_q:=min_{chi e chi_0} ert L^prime/L(1,chi)ert$, as $q o infty$ over the primes, where $L(s,chi)$ is the Dirichlet $L$-function attached to a non trivial Dirichlet character modulo $q$. Our main result shows that $m_q ll loglog q/sqrt{log q}$. We also compute $m_q$ for every odd prime $q$ up to $10^7$. As a consequence we numerically verified that for every odd prime $q$, $3 le q le 10^7$, we have $c_1/q< m_q< 5/sqrt{q}$, with $c_1=21/200$. In particular, this shows that $L^prime(1,chi) e 0$ for every non trivial Dirichlet character $chi$ mod $q$ where $3leq qleq 10^7$ is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the $m_q$-values, see Section 6. The programs used and the computational results described here are available at the following web address: \url{http://www.math.unipd.it/~languasc/smallvalues.html}.