Explicit upper bounds for |L(1,chi)| when chi is even

Let $\chi$ be a primitive Dirichlet character of conductor $q$ and let us denote by $L(s,\chi)$ the associated $L$-series. It is well known that there exists a constant $C$ such that $|L(1,\chi)|$ satisfies the following bound: $|L(1,\chi)|\leq \tfrac 12 \log q+C \qquad (q>1).$ Recall that $\chi$ is said to be even or odd according to whether $\chi(-1)=1$ or $\chi(-1)=-1$. It has been proven by Ramaré that $C=0$ is possible when $\chi$ is even and $C=0.7082$ when $\chi$ is odd. In the case $\chi (2)\neq 1$, Ramaré, following the work of Louboutin, has already proposed an explicit improvement of the bound above. In this talk, we examine the harder case $\chi(2)=1$. We present a method that leads to a better value of $C$ when $\chi$ is even, $\chi(2)=1$.