String Topology, Euler Class and TNCZ free loop fibrations

Let $M$ be a connected, closed oriented manifold. Let $\omega\in H^m(M)$ be its orientation class. Let $\chi(M)$ be its Euler characteristic. Consider the free loop fibration \Omega M\buildrel{i}\over\hookrightarrow LM\buildrel{ev}\over\twoheadrightarrow M. For any class $a\in H^*(LM)$ of positive degree, we prove that the cup product $\chi(M)a\cup ev^*(\omega)$ is null. In particular, if $i^*:H^*(LM;\mathbb{F}_p)\twoheadrightarrow H^*(\Omega M;\mathbb{F}_p)$ is onto then $\chi(M)$ is divisible by $p$ (or $M$ is a point).