Zeros of $L$-functions and large partial sums of Dirichlet coefficients

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