6. In the Cartesian coordinate system $x O y$, the parabola $\Gamma: y^{2}=2 p x(p>0)$ has a focus at $F$. A tangent line to $\Gamma$ is drawn through a point $P$ on $\Gamma$ different from $O$, intersecting the $y$-axis at point $Q$. If $|F P|=2,|F Q|=1$, then the dot product of vectors $\overrightarrow{O P}$ and $\overrightarrow{O Q}$ is $\qquad$ .
As shown in the figure, draw $P M$ perpendicular to the directrix of the parabola, intersecting the directrix at point $M$, then $\triangle P F M$ is an equilateral triangle with side length 2. $|O F|=\frac{1}{2} \Rightarrow p=1$, the equation of the parabola is $y^{2}=2 x$, thus $P\left(\frac{3}{2}, \sqrt{3}\right), Q\left(0, \frac{\sqrt{3}}{2}\right) \Rightarrow \overrightarrow{O P} \cdot \overrightarrow{O Q}=\frac{3}{2}$