CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Parity-forced trivial zeros

Statement

For primitive nonprincipal $χ$ of parity $a$ , $L (- a - 2 m, χ) = 0$ for every $m \geq 0$ . Thus $0$ occurs only in the even nonprincipal case; no zero at $0$ is asserted for the principal character modulo $1$ .

Facts & Assumptions

Given: A primitive nonprincipal $χ$ of parity $a$ .

[F1]

The completed function is entire for a primitive nonprincipal character (Analytic continuation of primitive Dirichlet L-functions).

[F2]

Gamma has simple poles at the nonpositive integers (Meromorphic continuation of Gamma).

Proof

technique · direct

1.1

At $s = - a - 2 m$ , $Γ ((s + a) / 2)$ has a pole by [F2].

F2given

2.1

The regularity in [F1] forces $L (s, χ)$ to cancel that pole. The principal exception is outside the hypothesis.

F1step 1.1∎

Depends on

Used by

Trivial zeros of the beta function Example

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andersen, sections 16.3.1-16.3.2 (standard reference, not scraped)