{
  "abstract": "Let $S_k$ be the space of cusp forms of weight $k$ for ${\\rm SL}_2(\\mathbb Z)$, with $\\dim S_k\\geq1$, and let $χ_p=\\left(\\frac{\\cdot}{p}\\right)$ for odd primes $p$. For fixed $s$ with ${\\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\\leq X$ satisfying $L(h,χ_p,s)=0$ tends to zero as $X\\to\\infty$, uniformly over nonzero $h\\in S_k$. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square $L$-values of the normalized Hecke eigenforms when ${\\rm Re}(s)>k$. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in $S_k$ for the kernels associated with twisted standard and symmetric square $L$-functions, with explicit basis criteria in specified right half-planes when $\\dim S_k=2$ or $3$. We show that almost every tuple of $\\dim S_k$ primes gives a basis of $S_k^{\\ast}$ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of $S_k$ obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11974",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianyu Ni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/064tqnaf0nbzd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11974",
  "fulltext_url": null,
  "id": "064tqnaf0nbzd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11974",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221183052",
  "pdf_url": "https://arxiv.org/pdf/2610.11974",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/064tqnaf0nbzd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11974",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11974"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions",
  "title_text": "Uniform non-vanishing of prime quadratic twists of standard and symmetric square L-functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/064tqnaf0nbzd",
  "venue": "arXiv",
  "versions": []
}