Limits of traces of singular moduli

Let $f$ and $g$ be weakly holomorphic modular functions on $\Gamma_0(N)$ with the trivial character. For an integer $d$, let $\Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $i\infty$ under the action of $\Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{\hat{H}(f)}(r)$ of the sum $H(f)(z) = \sum_{d>0}\Tr_d(f)e^{2\pi idz}$ is a special value of a regularized twisted $L$-function defined by $\Tr_d(f)$ for $d\leq0$. It is proved that the regularized $L$-function is meromorphic on $\mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{\hat{H}(f)}(r)=Q_{\hat{H}(g)}(r)$ for all $r$ equivalent to $i \infty$ under the action of $\Gamma_0(4N)$, then $\Tr_d(f)=\Tr_d(g)$ for all integers $d$.