  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 $d$.
