Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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Nonprincipal Dirichlet L-functions are nonzero at one

Statement

If χ≠χ0 is a Dirichlet character, then L(1,χ)≠0.

Facts & Assumptions

Given: A nonprincipal Dirichlet character χ.

[L1]

Nonreal Dirichlet characters satisfy L(1,χ)≠0 (A nonreal Dirichlet L-function is nonzero at one).

[L2]

Real nonprincipal Dirichlet characters satisfy L(1,χ)≠0 (A real nonprincipal Dirichlet L-function is nonzero at one).

Proof

technique · direct
1.1

Every Dirichlet character is either real or nonreal. If χ is nonreal, [L1] applies; if χ is real, then because it is also nonprincipal [L2] applies.

L1L2givencases
2.1

In both cases L(1,χ)≠0, so the theorem follows.

step 1.1cases-exhaustive∎

Depends on

Used by

Dependency tree · two levels

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Sources