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.
On the conjecture of birch and swinnerton-dyer.Inventiones mathematicae, 39(3):223–251, 1977
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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.