The linear independence of $1$, $\zeta (2)$, and $L(2,{\chi }_{-3})$

We prove the irrationality of the classical Dirichlet L-value $L(2,{\chi }_{-3})$. The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our work also establishes the $\mathbf{Q}$-linear independence of $1$, $\zeta (2)$, and $L(2,{\chi }_{-3})$. We also give a number of other applications of our method to other problems in irrationality.