Predicting root numbers with neural networks

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We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions.

• The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers.
• We show experimentally that these models achieve high accuracy by learning a combination of Mestre-Nagao type heuristics and murmurations, noting that the relative importance of these features varies with degree.
• The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time.