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We show that every Fricke-invariant meromorphic modular form for
whose divisor on
is defined over
and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight
. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of
-functions of certain weight
newforms. We also prove similar results for twisted Borcherds products.
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