---
title: "Low-lying zeroes of Maass form L-functions"
authors:
  - "Levent Alpoge"
  - "Steven J. Miller"
year: 2015
venue: "International Mathematics Research Notices"
arxiv: "1301.5702"
doi: "10.1093/imrn/rnu012"
source: "https://arxiv.org/abs/1301.5702"
topics:
  - "number-theory"
  - "modular-forms"
seed_rank: 1215
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  - title: "Maass waveforms and low-lying zeros"
    url: "https://arxiv.org/abs/1306.5886"
    year: 2013
    arxiv: "1306.5886"
    doi: "10.1007/978-3-319-22240-0_2"
see:
  []
---

# Low-lying zeroes of Maass form L-functions

## One-sentence takeaway

One-level density of level-1 Maass forms, as eigenvalues go to infinity, agrees with orthogonal type for test functions supported inside (-2+2/(2M+1), 2-2/(2M+1)), via Kuznetsov weights vanishing to order M at 0.

## Why it's lovely

The Maass sibling of Iwaniec-Luo-Sarnak's holomorphic-newform densities. Support past (-1,1) distinguishes the three orthogonal groups.

## Problem

Katz-Sarnak for GL2 Maass forms at the central point.

## Design

Smoothed Kuznetsov, vanishing-to-order-M weights, one-level density.

## Evidence

IMRN 2015, no. 10, 2678-2701. DOI 10.1093/imrn/rnu012. arXiv:1301.5702.

## Limitations

Level 1; support depends on M. See 1216 for the family-with-level variant.

## Implications for Broadside

Miller-collaboration analytic-NT card.

## Bottom line

Mint IMRN.

## Links

- arXiv: [1301.5702](https://arxiv.org/abs/1301.5702)
- DOI: [10.1093/imrn/rnu012](https://doi.org/10.1093/imrn/rnu012)