  Recent work has focused on the roots z of the Ehrhart polynomial of a lattice
polytope P. The case when Re(z) = -1/2 is of particular interest: these
polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such
polytopes when dim(P) <= 7. We also consider the "half-strip condition", where
all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds
for any reflexive polytope with dim(P) <= 5. We give an example of a
10-dimensional reflexive polytope which violates the half-strip condition, thus
improving on an example by Ohsugi--Shibata in dimension 34.
