# Properties

 Degree 2 Conductor 229 Sign $-0.462 + 0.886i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.23i·2-s + 3-s − 3.00·4-s + 3·5-s − 2.23i·6-s + 2.23i·8-s − 2·9-s − 6.70i·10-s + 3·11-s − 3.00·12-s + 3·15-s − 0.999·16-s − 3·17-s + 4.47i·18-s − 19-s − 9.00·20-s + ⋯
 L(s)  = 1 − 1.58i·2-s + 0.577·3-s − 1.50·4-s + 1.34·5-s − 0.912i·6-s + 0.790i·8-s − 0.666·9-s − 2.12i·10-s + 0.904·11-s − 0.866·12-s + 0.774·15-s − 0.249·16-s − 0.727·17-s + 1.05i·18-s − 0.229·19-s − 2.01·20-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$229$$ $$\varepsilon$$ = $-0.462 + 0.886i$ motivic weight = $$1$$ character : $\chi_{229} (228, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 229,\ (\ :1/2),\ -0.462 + 0.886i)$ $L(1)$ $\approx$ $0.827711 - 1.36545i$ $L(\frac12)$ $\approx$ $0.827711 - 1.36545i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 229$, $$F_p$$ is a polynomial of degree 2. If $p = 229$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad229 $$1 + (-7 + 13.4i)T$$
good2 $$1 + 2.23iT - 2T^{2}$$
3 $$1 - T + 3T^{2}$$
5 $$1 - 3T + 5T^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 - 3T + 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 3T + 17T^{2}$$
19 $$1 + T + 19T^{2}$$
23 $$1 - 4.47iT - 23T^{2}$$
29 $$1 - 4.47iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 - 4.47iT - 41T^{2}$$
43 $$1 + T + 43T^{2}$$
47 $$1 + 8.94iT - 47T^{2}$$
53 $$1 - 6T + 53T^{2}$$
59 $$1 + 8.94iT - 59T^{2}$$
61 $$1 - 5T + 61T^{2}$$
67 $$1 + 13.4iT - 67T^{2}$$
71 $$1 + 15T + 71T^{2}$$
73 $$1 - 13.4iT - 73T^{2}$$
79 $$1 - 13.4iT - 79T^{2}$$
83 $$1 + 9T + 83T^{2}$$
89 $$1 - 17.8iT - 89T^{2}$$
97 $$1 + 7T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}