# E069: L(1,χ): slow convergence and smoothing.

```{figure} ../_static/experiments/e069_hero.png
:width: 80%
:alt: Preview figure for E069
```

```{figure} ../_static/experiments/e069_hero_2.png
:width: 80%
:alt: Preview figure for E069
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `dirichlet-characters`, `l-functions`

## Highlights
- Computes truncated Dirichlet L-series for a nontrivial character.
- Compares series truncation against Euler-product truncation (where meaningful).
- Shows convergence behavior (fast for s>1, slow at s=1).

## What this experiment does

At s=1, the Dirichlet series for L(1,χ) converges very slowly. For the nontrivial character modulo 4:

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

## Outputs

This experiment writes into `out/e069/`:

- `figures/fig_01_l1_partial_sums.png`
- `figures/fig_02_l1_smoothed.png`

## How to run

```bash
make run EXP=e069
```

## Notes

- The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.
- Prime-race plots depend on the chosen sampling of `x` (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

## Published run snapshot

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

```{include} ../reports/e069.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e069.json
:language: json
```

:::

## References

{cite:t}`apostol1976introanalyticnumbertheory, montgomeryvaughan2006multiplicativenumbertheoryi`

## Related experiments

- {doc}`e068` (Dirichlet L(s,χ): series vs. Euler product (partial approximations).)
- {doc}`e107` (E107: Conductor: primitive vs. induced characters)
- {doc}`e122` (E122: Character averages over primes)
- {doc}`e110` (E110: Dirichlet L-series partial sums at s=1 and s=1/2)
- {doc}`e111` (E111: Euler product vs. Dirichlet series for L(s,χ))