Like Emil noted in the comments, the question is equivalent to whether $0\in\mathrm{Conv}(\mathbb{Z}^d\setminus\{0\})$. To see that it is, note that it is the average of $(d-1,-1,-1,\ldots,-1)$ and its permutations.

Like Emil noted in the comments, the question is equivalent to whether $0\in\mathrm{Conv}(\{\alpha\in\mathbb{Z}^d: \sum_{i=1}^{d}\alpha_i=0\}\setminus\{0\})$. To see that it is, note that it is the average of $(d-1,-1,-1,\ldots,-1)$ and its permutations.