Machine learning the vanishing order of rational L-functions

TL;DR

This paper applies data science techniques to analyze the vanishing order of rational L-functions, revealing patterns and enabling accurate predictions through PCA, LDA, and neural networks.

Contribution

It introduces a data-driven approach to classify and predict the vanishing order of rational L-functions using machine learning methods.

Findings

PCA clusters L-functions by vanishing order

LDA and neural networks predict vanishing order accurately

Observed murmuration-like patterns in dataset averages

Abstract

In this paper, we study the vanishing order of rational $L$-functions from a data scientific perspective. Each $L$-function is represented in our data by finitely many Dirichlet coefficients, the normalisation of which depends on the context. We observe murmuration-like patterns in averages across our dataset, find that PCA clusters rational $L$-functions by their vanishing order, and record that LDA and neural networks may accurately predict this quantity.