The landscape of L-functions: degree 3 and conductor 1
Bibliographic record
Abstract
We extend previous lists by numerically computing approximations to many L-functions of degree d = 3 d=3 , conductor N = 1 N=1 , and small spectral parameters. We sketch how previous arguments extend to show that for very small spectral parameters there are no such L-functions. Using the case ( d , N ) = ( 3 , 1 ) (d,N) = (3,1) as a guide, we explain how the set of all L-functions with any fixed invariants ( d , N ) (d,N) can be viewed as a landscape of points in a ( d − 1 ) (d-1) -dimensional Euclidean space. We use Plancherel measure to identify the expected density of points for large spectral parameters for general ( d , N ) (d,N) . The points from our data are close to the origin and we find that they have smaller density.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.023 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.006 | 0.010 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.023 | 0.007 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".