General form of the function $f(\mathbb{Q})$ using cylindrically static spacetime

We find an exact static solution in four dimensions to the field equations of the $f(\mathbb{Q})$ gravity by using a cylindrically static spacetime with two different ansatz, $\nu(r)$ and $\mu(r)$. This solution is derived without imposing any conditions on $f(\mathbb{Q})$. The black hole solution involves four constants: $c_1$, $c_2$, $c_3$, and $c_4$. Among these, $c_1$ is linked to the cosmological constant, $c_2$ to the black hole's mass, while $c_3$ and $c_4$ are responsible for the deviation of the solution from the linear form of $f(\mathbb{Q})$. We demonstrate how the analytical function $f(\mathbb{Q})$ relies on $c_3$. When $c_3$ is zero, $f(\mathbb{Q})$ becomes a constant function, leading to the non-metricity case. We investigate the singularity of this solution and show that the Kretschmann invariant has a much milder singularity compared to the non-metricity case. We produce a black hole that rotates with non-vanishing values of $\mathbb{Q}$ and $f(\mathbb{Q})$ by using a coordinate transformation. Then, we analyze the laws of thermodynamics to determine the physical characteristics of this black hole solution and demonstrate that it is locally thermodynamically stable.