The displayed analytic rank $r$ is an upper bound on the true analytic rank that is believed to be tight; it is known that there are $r$ (but not $r+1$) zeros within a region of $s=1/2$ (in the analytic normalization) of radius equal to the error implied by the displayed precision of the zeros). In particular, any displayed analytic rank $r=0$ must be correct.
For self-dual L-functions the parity of the analytic rank is determined by the sign $\varepsilon=\pm 1$ of the functional equation that all modular L-functions are known to satisfy. The parity of the displayed analytic rank is always consistent with the sign, thus for self-dual modular L-functions any displayed analytic rank $r\le 1$ must be correct.