A quadratic critical-value conjecture for the fifth Bessel moment

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 and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $⟨f,f{⟩}_{60}=\frac{3(5-\sqrt{5})}{2{\pi }^4}|L(f,2){|}^2$, the coefficient-conjugation relation $L(f^{\sigma },2)=\kappa L(f,2)$ with explicit $\kappa \in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L({\chi }_{-4}{\mathrm{Sym}}^{2}f,2)=\sqrt{15}{\pi }^2⟨f,f{⟩}_{60}$, together with $L({\chi }_{-4}{\mathrm{Sym}}^{2}f,3)={\pi }^4⟨f,f{⟩}_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}}=\frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2){|}^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}}={\pi }^{2}L({\chi }_{-4}{\mathrm{Sym}}^{2}f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}}={\pi }^2{D}_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.