Universal dominance conjecture for the residue class $-1 \pmod N$

Let $N\ge 3$, and for each reduced residue class $a\in(\mathbb{Z}/N\mathbb{Z})^\times$ let $\chi_{1,a}$ denote the character associated with the fine-structure bias of the class $a$. Consider the quantity $\log L(1,\chi_{1,a})$ over all non-principal reduced residue classes.

Universal dominance conjecture. For any modulus $N\ge 3$, the quantity $\log L(1,\chi_{1,a})$ achieves its unique absolute minimum over all non-principal residue classes $a\in(\mathbb{Z}/N\mathbb{Z})^\times\setminus\{1\}$ at $a\equiv -1\pmod N$. Consequently, $-1\pmod N$ exhibits the maximal fine-structure prime bias across all reduced residue classes.

The conjecture formalizes the claimed universal dominance of the residue class $-1\pmod N$ in the fine-structure hierarchy: smaller values of $\log L(1,\chi_{1,a})$ correspond to stronger prime bias. The source presents exact spectral calculations and numerical examples for several moduli, but does not establish the assertion for every $N\ge 3$.