In 1965, Birch and Swinnerton-Dyer formulated a conjecture on the Mordell-Weil rank $r$ of elliptic curves which also implies the convergence of the Nagao-Mestre sum. We show that if the Nagao-Mestre sum converges, then the limit equals $-r+1/2$, and study the connections to the Riemann hypothesis for E. We also relate this to Nagao's conjecture for elliptic curves. Furthermore, we discuss a generalization of the above results for the Selberg classes and hence (conjecturally) for larger classes of $L$-functions.