The authors construct a weak Maass form omega_{k+1,D} related to the divisor of Zagier's f_{k,D} and show its generating function yields a new theta lift.
Central $L$-values of newforms and local polynomials
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abstract
In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.
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On a Divisor Modular Form and a Theta Lift
The authors construct a weak Maass form omega_{k+1,D} related to the divisor of Zagier's f_{k,D} and show its generating function yields a new theta lift.