  We define the property (D) for nontrivial knots. We show that the fundamental
group of the manifold obtained by Dehn surgery on a knot $K$ with property (D)
with slope $\frac{p}{q}\ge 2g(K)-1$ is not left orderable. By making full use
of the fixed point method, we prove that (1) nontrivial knots which are
closures of positive $1$-bridge braids have property (D); (2) L-space satellite
knots, with positive $1$-bridge braid patterns, and companion with property
(D), have property (D); (3) the fundamental group of the manifold obtained by
Dehn filling on $v2503$ is not left orderable. Additionally, we prove that
L-space twisted torus knots of form $T_{p,kp\pm 1}^{l,m}$ are closures of
positive $1$-bridge braids.
