We show that in Weil's converse theorem the functional equations of multiplicative twists for at least the first $\sqrt{\frac{p-24}{3}}$ moduli are needed in order to prove the modularity for $\Gamma_0(p)$.
Character sums, reciprocity and functional equations
Functional Equations of $L$-Functions for Symmetric Products of the Kloosterman Sheaf
Functional equations of the dilogarithm in motivic cohomology
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