LI conjecture on exact symplectic central vanishing order

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Let $W(\chi)$ be the root number of the Artin $L$-function attached to $\chi$, and let $\chi_0$ denote the trivial character. LI conjecture. The LI− conjecture is true, and if $\chi$ is a symplectic irreducible character of $\operatorname{Gal}(L_0/\mathbb Q)$, then

$$\operatorname{ord}_{s=1/2}L(s,\chi,L_0/\mathbb Q)=\frac{1-W(\chi)}{2}.$$

For symplectic characters the root number forces central vanishing when it is $-1$; the supplied source gives no resolution of this exact-order assertion.