Generate a simulated data matrix for a given convariance

Question

I have the covariance matrix, let's call it Z of size pxp.

I want to come up with a matrix X of generated data and of size nxp that satisfies three conditions:

1. The columns are normally distributed.
2. The columns are centered at 1.
3. The matrix of covariance for X is Z.

I am looking for both mathematichal and algorithmic way to do it.

Thanks.