For the ordinary least square (OLS) estimates of regression ($\vec{y} =\mathbf{X} \cdot \vec{\beta} + \vec{\epsilon}$) to be unbiased (without considering the efficiency), which one of the three conditions is required (sufficient):
$$\mathbb{E}(\vec{\epsilon}) = \vec{0} \tag{Assumption 1}$$
$$\mathbb{E}(\vec{\epsilon} | \mathbf{X}) = \vec{0} \tag{Assumption 2}$$
$$\mathbb{E}(\mathbf{X}^T \cdot \vec{\epsilon}) = 0 \tag{Assumption 3}$$$$\mathbb{E}(\mathbf{X}^T \cdot \vec{\epsilon}) = \vec{0} \tag{Assumption 3}$$
I came to the first one when deriving by myself:
The expectation of $\hat{\vec{\beta}}$ is:
$$ \begin{align*} \mathbb{E}(\hat{\vec{\beta}}) & = \mathbb{E}[\vec{\beta} + (\mathbf{X}^T \cdot \mathbf{X})^{-1} \cdot \mathbf{X}^T \cdot\vec{\epsilon}] \\ & = \mathbb{E}(\vec{\beta}) + \mathbb{E}[(\mathbf{X}^T \cdot \mathbf{X})^{-1} \cdot \mathbf{X}^T \cdot\vec{\epsilon}] \\ & as\ \vec{\beta} \ is \ constant\ and\ \mathbf{X}\ is\ also\ constant\\ & = \vec{\beta} + (\mathbf{X}^T \cdot \mathbf{X})^{-1} \cdot \mathbf{X}^T \cdot \mathbb{E}(\vec{\epsilon}) \\ \end{align*} $$
But some materials in econometrics stated the Assumption 2 or Assumption 3. I am confused as $\mathbf{X}$ is considered observed or fixed in OLS, conditioned expectation in Assumption 2 doesn't seem to make sense. And Assumption 3 should easily be simplified to Assumption 1 given $\mathbf{X}$ fixed.