The analytic rank of an elliptic curve $E$ is the analytic rank of its L-function $L(E,s)$. The weak form of the BSD conjecture implies that the analytic rank is equal to the rank of the Mordell-Weil group of $E$.
For elliptic curves $E$ over $\Q$, it is known that $L(E,s)$ satisfies the Hasse-Weil conjecture, and hence that the parity of the analytic rank is always compatible with the sign of the functional equation.
In general, analytic ranks stored in the LMFDB are only upper bounds on the true analytic rank (they could be incorrect if $L(E,s)$ had a zero very close to but not on the central point). For elliptic curves over $\Q$ of analytic rank less than 2 this upper bound is necessarily tight, due to parity; for analytic ranks $2$ and $3$ is also tight due to results of Kolyvagin; Murty and Murty; Bump, Friedberg and Hoffstein; Coates and Wiles; Gross and Zagier which together say that when the analytic rank is $0$ or $1$ then it equals the Mordell-Weil rank.
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- Last edited by John Cremona on 2020-12-15 07:19:31