Kicking one Dirichlet coefficient of the Davenport–Heilbronn function: the kill cost of an off-line pair, and what the rest of the zeros do

We change one Dirichlet coefficient of the Davenport–Heilbronn function, fε(s) = f(s) + ε m−s with m = 2 or 7 (the two smallest primes in the support of its multiplicativity defect) and ε real. The functional equation is lost, and every simple zero moves at first order with the known velocity −m−ρ/f′(ρ). Four items were registered before computing: the velocity at the first off-line pair, 0.8085 + 85.699i, checked against Arb root tracking; the cost of a kick that brings that zero within 0.01 of the line (the pair approaches the line only for ε < 0); the number of zeros more than 0.01 off the line at that kick, counted by the argument principle on [−2, 3] × [12, 2000] and certified zero by zero; and the shift of the mean real part. Killing one pair by a kick scatters the whole spectrum by a comparable amount: the kick does not restore the line, it removes it. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, and the scripts, registered predictions and result files it cites.