  Let $L(s,\pi)=\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s}$ be an $L$-function
that satisfies a weak form of the generalized Ramanujan conjecture. We prove
that large partial sums of $\lambda_{\pi}(n)$ strongly repel the low-lying
zeros of $L(s,\pi)$ away from the critical line. Our results extend and
quantitatively improve preceding work of Granville and Soundararajan.
