Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere:

Let $E$ be an elliptic curve, and $f(q)$ its associated modular form. Is it known whether along the positive imaginary axis, the number of roots of $f(q)$ is equal to the order of $L(E,1)$, i.e. the analytic rank?

I've checked a few cases on LMFDB, and the plots of $f\left(e^{-2\pi x}\right)$ along $x>0$ seem to show the same number of roots as the analytic rank of the L-function (or equivalently the algebraic rank of the elliptic curve assuming BSD). One requires several thousand terms of the q-expansion when the conductor is large to have sufficient convergence for all roots to appear. I am curious to know if this has been conjectured before (say as part of the Langlands program) or if it is a known result, or perhaps is actually false?