Machine learning the vanishing order of rational L-functions

TL;DR

The paper tackles predicting the vanishing order $r=\operatorname{ord}_{s=(w+1)/2} L(s)$ of rational $L$-functions from Dirichlet coefficients $\{a_p\}$ using machine learning. It builds the RAT/PRAT datasets, defines normalized feature vectors $\overline{a}_p$ and $v(L)\in\mathbb{R}^{168}$, and applies PCA, LDA, and CNNs to learn $r$, achieving high accuracy and demonstrating transfer learning across subdatasets like ECNF and G2Q. The results show strong separability of vanishing orders and reveal that murmuration-like coefficient patterns encode arithmetic information related to the central value, with connections to Mestre–Nagao sums. These findings offer data-driven insights into arithmetic invariants and suggest avenues for further mathematical and ML investigations across broader conductor ranges.

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.

Figures (13)

• Figure 2.1: UpSet plot for RAT. Each row corresponds to the named subset, and each column corresponds to an intersection of subsets. A black circle is used to indicate that a subset is involved in a particular intersection (the vertical lines simply indicate the orientation of the plot). The horizontal bars show the number of datapoints in each subset (row), and a vertical bar above shows the size of an exclusive intersection, i.e., it represents the number of elements unique to that specific intersection.
• Figure 2.2: Average value of $\widetilde{a}_p$ for primitive rational $L$-functions with specified vanishing order, excluding the $9$$L$-functions with vanishing order $4$.
• Figure 2.3: Average value of $\widetilde{a}_p$ over $L$-functions with specified vanishing order in $\texttt{PRAT}^{\star}$.
• Figure 2.4: Average value of $\widetilde{a}_p$ over $L$-functions with specified vanishing order in (left) ECNF and (right) G2Q excluding the $27$$L$-functions with vanishing order $3$ for ECNF.
• Figure 2.5: Venn diagram for ECNF.