Let us consider the one-dimensional case. By the Lévy-Khintchine formula, we have$$E(e^{i\theta X_1})= e^{-\phi(\theta)},$$ where
$$\phi(\theta)=\int_{R}(1-e^{i\theta x}+i\theta x 1_{|x|<1})\nu(dx).$$
Then
$$E(iX_1e^{i\theta X_1})= e^{-\phi(\theta)}\int_{R}(ixe^{i\theta x}-i x 1_{|x|<1})\nu(dx).$$
Letting $\theta\to 0$ we see
$$E(X_1)= \int_{R}(x-x 1_{|x|<1})\nu(dx)= \int_{|x|\geq 1}x \nu(dx).$$
Then we see that the expectation is only determined by the “big jumps term” (provided there is neither a drift term nor a Brownian motion term).