The conductor is claimed to be the minimal integer that can appear in a degree-two functional equation or analytic rank bound for an elliptic curve over Q.
Random maximal isotropic subspaces and selmer groups.Journal of the AMS, 25(1):245–269, 2012
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On the Minimality of the Conductor for Elliptic Curve $L$-Functions
The conductor is claimed to be the minimal integer that can appear in a degree-two functional equation or analytic rank bound for an elliptic curve over Q.