# Properties

 Label 2-230-5.4-c1-0-6 Degree $2$ Conductor $230$ Sign $1$ Analytic cond. $1.83655$ Root an. cond. $1.35519$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 0.618i·3-s − 4-s + 2.23·5-s + 0.618·6-s − 4.85i·7-s − i·8-s + 2.61·9-s + 2.23i·10-s − 3.38·11-s + 0.618i·12-s + 0.381i·13-s + 4.85·14-s − 1.38i·15-s + 16-s + 5.85i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.356i·3-s − 0.5·4-s + 0.999·5-s + 0.252·6-s − 1.83i·7-s − 0.353i·8-s + 0.872·9-s + 0.707i·10-s − 1.01·11-s + 0.178i·12-s + 0.105i·13-s + 1.29·14-s − 0.356i·15-s + 0.250·16-s + 1.41i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$1.83655$$ Root analytic conductor: $$1.35519$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{230} (139, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.36608$$ $$L(\frac12)$$ $$\approx$$ $$1.36608$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
5 $$1 - 2.23T$$
23 $$1 - iT$$
good3 $$1 + 0.618iT - 3T^{2}$$
7 $$1 + 4.85iT - 7T^{2}$$
11 $$1 + 3.38T + 11T^{2}$$
13 $$1 - 0.381iT - 13T^{2}$$
17 $$1 - 5.85iT - 17T^{2}$$
19 $$1 - 6.85T + 19T^{2}$$
29 $$1 + 3.70T + 29T^{2}$$
31 $$1 + 8.85T + 31T^{2}$$
37 $$1 + 3.70iT - 37T^{2}$$
41 $$1 + 3.38T + 41T^{2}$$
43 $$1 - 6.76iT - 43T^{2}$$
47 $$1 - 11.7iT - 47T^{2}$$
53 $$1 + 2iT - 53T^{2}$$
59 $$1 - 6T + 59T^{2}$$
61 $$1 + 3.85T + 61T^{2}$$
67 $$1 + 0.763iT - 67T^{2}$$
71 $$1 - 2.61T + 71T^{2}$$
73 $$1 + 7.52iT - 73T^{2}$$
79 $$1 + 5.70T + 79T^{2}$$
83 $$1 + 5.70iT - 83T^{2}$$
89 $$1 - 9.70T + 89T^{2}$$
97 $$1 - 16.0iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$