3. If $z \in C$, $\arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, then the value of $z$ is $\qquad$
3. As shown in the right figure, using the geometric meaning of complex numbers.
$z^{2}+4$ can be represented as $\overrightarrow{O A}$,
$z^{2}-4$ can be represented as $\overrightarrow{O B}$,
$z^{2}$ can be represented as $\overrightarrow{O C}$,

and $\angle A O B=\frac{\pi}{2}$, and $C$ is the midpoint of segment
$A B$,
$\therefore|O C|=4$, i.e.
$$
\begin{array}{l}
\left|z^{2}\right|=4, \arg z^{2}=\frac{5 \pi}{6}-\frac{\pi}{6}=\frac{2 \pi}{3} . \\
\therefore \quad z^{2}= 4\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right), \\
\text { hence } \quad z= \pm 2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)= \pm 2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right) \\
= \pm(1+\sqrt{3} i) .
\end{array}
$$