Conductor Ordering of Li Coefficients Across Dirichlet Characters

The Li coefficient literature — from Li (1997) through Bombieri–Lagarias (1999) and Lagarias (2007) — treats even and odd Dirichlet characters uniformly. We show that this uniformity obscures a permanent spectral hierarchy. Restricting to even primitive characters (χ(-1) = +1), the natural sector containing all spectral data for ζ, reveals an exact conductor ordering of the first Li coefficients λ₁(χ): the archimedean term ½ log f dominates, the arithmetic correction Re(L'/L(1,χ*)) is bounded, and the ordering is stable through the primorial tower. The gap between conductor 1 (the Riemann zeta function) and conductor 5 (the Legendre symbol mod 5) is δ = 0.05518…, a fixed arithmetic constant.

The parity filter is the key methodological step: the digamma shift ln 2 ≈ 0.693 between even and odd characters exceeds δ by a factor of 12, interleaving the two families at higher conductors and burying the hierarchy when both are included. The same conductor ordering appears independently in function field Newton polygons (against Weil's theorem) and in Selberg scattering integrals for Γ₀(5). Five open problems are identified.