A zero density estimate and fractional imaginary parts of zeros for $\mathrm{GL}_2$ $L$-functions

We prove an analogue of Selberg’s zero density estimate for ζ(s) that holds for any GL2 L-function. We use this estimate to study the distribution of the vector of fractional parts of γα, where α ∈ R is fixed and γ varies over the imaginary parts of the nontrivial zeros of a GL2 L-function.

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