The generalised error distribution (sometimes called the generalised normal distribution) is a generalisation of the normal distribution allowing variation of the kurtosis away from mesokurtosis. Like the normal distribution, the generalised error distribution is symmetric, and therefore unskewed, but it allows variation of the kurtosis over a wide range of values. Given the shape parameter $\beta >0$, the standardised form of the generalised error distribution has the density kernel:
$$\text{GenErr}(z|\beta) \propto \exp \Bigg( -\bigg( \frac{\Gamma(3/\beta)}{\Gamma(1/\beta)} \bigg)^{\beta/2} \cdot |z|^\beta \Bigg) \quad \quad \quad \text{for } z \in \mathbb{R}.$$
The kurtosis of the distribution is a continuous monotonically decreasing function of the parameter $\beta$, and it can take values above the lower bound $\lim_{\beta \rightarrow \infty} \mathbb{Kurt}(Z) = 3/\sqrt{5} \approx 1.341641$ (which is not far above unit kurtosis, which is the lowest kurtosis of any distribution). In the special case where $\beta = 2$ the generalised error distribution reduces to the standard normal distribution.
My Question: What is the simplest method of generating this random variable?