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Let~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that thespecial $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for theratio of the special $L$-value to the real period of $A$ that expresses thisratio as a rational number. We extract an integer factor from the numerator ofthis formula; this factor is non-trivial in general and is related to certaincongruences of $f$ with eigenforms of positive analytic rank. We use thetechniques of visibility to show that, under certain hypotheses (which includesthe first part of the Birch and Swinnerton-Dyer conjecture on rank), if an oddprime $q$ divides this factor, then $q$ divides either the order of theShafarevich-Tate group or the order of a component group of $A$. Suppose $p$ isan odd prime such that $p^2$ does not divide $N$, $p$ does not divide the orderof the rational torsion subgroup of $A$, and $f$ is congruent modulo a primeideal over $p$ to an eigenform whose associated abelian variety has positiveMordell-Weil rank. Then we show that $p$ divides the factor mentioned above; inparticular, $p$ divides the numerator of the ratio of the special $L$-value tothe real period of $A$. Both of these results are as implied by the second partof the Birch and Swinnerton-Dyer conjecture, and thus provide theoreticalevidence towards the conjecture.

A visible factor of the special L-value