# E111: Euler product vs. Dirichlet series for L(s,χ)

```{figure} ../_static/experiments/e111_hero.png
:alt: E111 hero
:class: experiment-hero

E111: Euler product vs. Dirichlet series for L(s,χ)
```

**Tags:** `number-theory`, `l-functions`, `model-checking`, `visualization`, `dirichlet-series`, `numerics`

## Highlights

- Focused numeric experiment with a small set of figures.
- Parameters saved to `params.json` for reproducibility.
- Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

## What is computed

- Compare truncated Euler products and truncated series for L(s,χ).

## Notes

- This page summarizes the intent; see the generated `report.md` in `out/` for concrete outputs.

## Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

```{include} ../reports/e111.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e111.json
:language: json
```

:::

## References

- See `docs/background/dirichlet-l-functions.md` and `docs/background/dirichlet-convolution.md`.

## Related experiments

- {doc}`e068` (Dirichlet L(s,χ): series vs. Euler product (partial approximations).)
- {doc}`e110` (E110: Dirichlet L-series partial sums at s=1 and s=1/2)
- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)
- {doc}`e083` (E083: Series vs. Euler product (ζ))
- {doc}`e091` (E091: Partial Euler products on the critical line)