The eight characters of the modulo-30 wheel:

The eight vectors of the modulo-30 wheel form the group W = (Z/30Z)× ∼= C2 × C4.This group has exactly three subgroups of index two, the kernels of the three real charactersχ3, χ5, χ15. Each kernel has a meaning on the wheel. ker χ5 = {11, 19, 29, 31} consists ofthe vectors whose squares lie on the eighth vector V31; the other half, {7, 13, 17, 23}, hasits squares on the fifth vector V19 and holds no square at all. ker χ3 = {7, 13, 19, 31} is thecycle of V31 under the wave of 7, and ker χ15 = {17, 19, 23, 31} is the cycle of V31 under thewave of 17. A semiprime pq lies in a kernel exactly when χ(p) = χ(q). We test the zerosof ζ(s) and of L(s, χ) on the semiprimes of these halves with the spectral method of [3],applied to the weighted count Pln p ln q up to 5 × 107. On one vector the zeta zeros arehidden among the zeros of the eight L-functions modulo 30: one of the ten strongest peakslies on a zeta zero, no more than chance. For each of the three characters, the sum of thetwo halves carries only the zeta zeros (ten of the ten strongest peaks, within 0.07). Thedifference carries only the zeros of L(s, χ): ten of ten in each case, within 0.035 for χ3, 0.04for χ5 and 0.045 for χ15, against chance probabilities of 2 × 10−7, 10−6 and 5 × 10−5. Thethree halves thus separate the four spectra of the real characters exactly. The four complexcharacters of order 4 cannot be separated by halves, but they are separated by quarter turns,i.e. by the weights 1, i, −1, −i on the vectors. In the wave of 7 the twelve strongest peaksof each complex signal, at positive and negative frequencies, lie on zeros of L(s, ψ5) and ofL(s, ψ15). Finally, the wave of p multiplies the term of a zero ρ (γ = ℑρ) by χ(p) p−ρ. Thephase is the angle of deviation arg χ(p) plus the shift −γ ln p, and the modulus is p−ℜρ. Overthe waves of the 22 primes 7 ≤ p ≤ 97, the phase follows this law within 4.2◦. Without theangle arg χ(p) the error reaches 180◦. The ordinates γ are recovered from the dependenceon p alone within 0.035, and ℜρ from the decay of the modulus as 0.47–0.52. The resultsconfirm classical explicit formulas and make no claim about the Riemann hypothesis.