Classification of norm-Euclidean imaginary quadratic integer rings

Statement

The following are the precise values of $D<0$, $D$ square-free for which the imaginary quadratic integer ring corresponding to $D$ is a norm-Euclidean ring of integers.

$\{-1,-2,-3,-7,-11\}$.

Note that for $D\equiv 2\mod 4$ or $D\equiv 3\mod 4$, the quadratic integer ring is given by $\mathbb {Z} [{\sqrt {-D}}]$, while for $D\equiv 1\mod 4$, the quadratic integer ring is given by $\mathbb {Z} \left[{\frac {1+{\sqrt {D}}}{2}}\right]$.