Effective Determination of Primitive Dirichlet Characters by Their Zeros up to Polylogarithmic Height

This paper establishes an effective spectral distinction principle for primitive Dirichlet characters. Let (χ_1) and (χ_2) be distinct primitive Dirichlet characters of conductors at most (Q) and of orders dividing (d). Under the Riemann Hypothesis for the two associated Dirichlet (L)-functions and for the primitive character induced by (χ_1χ_2̄), the paper proves that their signed multisets of nontrivial zero ordinates must differ below an effectively computable height of polylogarithmic size in (Q). The proof combines the Guinand–Weil explicit formula with a band-limited test function whose Fourier transform isolates a single distinguishing prime power while annihilating the conductor term and the reflected prime-power channel. A quantitative least-distinguishing-prime-power estimate then converts arithmetic disagreement between the characters into disagreement between their finite zero data. A stable version is also developed for zero ordinates known only within a prescribed tolerance. In addition, an unconditional polynomial-height distinction theorem is obtained from character-sum bounds. The results quantify the information carried by finite zero data; they do not prove the Riemann Hypothesis or the Generalized Riemann Hypothesis.