We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $x^{″}+f(t,x)=sp(t)$. We weaken the usual requirement on the sublinearity of perturbations when $|x|$ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $f$ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.