Let $\chi_{k}(n)$ denote the Kronecker symbol $(k|n)$ and $\mathbb{Z}_{k}^{\times}$ the group of units modulo $|k|$. Under which assumptions on $k \in \mathbb{Z} \setminus \{0 \}$ can one conclude the following?
$\sum_{d \in \mathbb{Z}_{k}^{\times}} \chi_{k}(d) = 0$;
$\left| \sum_{d \in \mathbb{Z}_{k}^{\times}} d \chi_{k}(d) \right| \geq |k|$; and,
$k$ divides $\left| \sum_{d \in \mathbb{Z}_{k}^{\times}} d \chi_{k}(d) \right|$.
Clearly, $3.$ implies $2.$ if the sum is non-zero.
NB: This is not a homework problem.
Update: If $k > 0$ is congruent to $0$ or $3$ modulo $4$, then $\sum_{d \in \mathbb{Z}_{k}^{\times}} \chi_{-k}(d) = 0$. However, this particular sum also vanishes if $k$ is one of the following: ${18, 25, 126, 150, 162, 234, 242, \dots}$ Is this sequence known?