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arXiv · 2609.22318

A quadratic critical-value conjecture for the fifth Bessel moment

Jonas Matuzas
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Abstract

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5 dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real part. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove two exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$ and the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$. We also record the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ between the even and odd determinants of Lim, Tu and Yu, as a consequence of Chuang's period formulas, and distinguish their symmetric-square conjecture for $D_{5,\mathrm{odd}}$ from the individual-period formula proposed here. The Bessel-to-modular equality remains conjectural. Verification programs are included as ancillary files.

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Jonas Matuzas. 2026-09-15. A quadratic critical-value conjecture for the fifth Bessel moment. https://arxiv.org/abs/2609.22318

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