# Properties

 Label 2-230-5.3-c2-0-17 Degree $2$ Conductor $230$ Sign $-0.946 + 0.323i$ Analytic cond. $6.26704$ Root an. cond. $2.50340$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − i)2-s + (−1.36 − 1.36i)3-s − 2i·4-s + (−3.17 + 3.86i)5-s − 2.72·6-s + (7.21 − 7.21i)7-s + (−2 − 2i)8-s − 5.29i·9-s + (0.691 + 7.03i)10-s − 14.8·11-s + (−2.72 + 2.72i)12-s + (−4.91 − 4.91i)13-s − 14.4i·14-s + (9.57 − 0.941i)15-s − 4·16-s + (−17.8 + 17.8i)17-s + ⋯
 L(s)  = 1 + (0.5 − 0.5i)2-s + (−0.453 − 0.453i)3-s − 0.5i·4-s + (−0.634 + 0.772i)5-s − 0.453·6-s + (1.03 − 1.03i)7-s + (−0.250 − 0.250i)8-s − 0.588i·9-s + (0.0691 + 0.703i)10-s − 1.34·11-s + (−0.226 + 0.226i)12-s + (−0.377 − 0.377i)13-s − 1.03i·14-s + (0.638 − 0.0627i)15-s − 0.250·16-s + (−1.04 + 1.04i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $-0.946 + 0.323i$ Analytic conductor: $$6.26704$$ Root analytic conductor: $$2.50340$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{230} (93, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :1),\ -0.946 + 0.323i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.181275 - 1.08924i$$ $$L(\frac12)$$ $$\approx$$ $$0.181275 - 1.08924i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 + i)T$$
5 $$1 + (3.17 - 3.86i)T$$
23 $$1 + (3.39 + 3.39i)T$$
good3 $$1 + (1.36 + 1.36i)T + 9iT^{2}$$
7 $$1 + (-7.21 + 7.21i)T - 49iT^{2}$$
11 $$1 + 14.8T + 121T^{2}$$
13 $$1 + (4.91 + 4.91i)T + 169iT^{2}$$
17 $$1 + (17.8 - 17.8i)T - 289iT^{2}$$
19 $$1 + 29.6iT - 361T^{2}$$
29 $$1 - 11.0iT - 841T^{2}$$
31 $$1 - 42.3T + 961T^{2}$$
37 $$1 + (-14.7 + 14.7i)T - 1.36e3iT^{2}$$
41 $$1 - 23.9T + 1.68e3T^{2}$$
43 $$1 + (-0.419 - 0.419i)T + 1.84e3iT^{2}$$
47 $$1 + (-53.0 + 53.0i)T - 2.20e3iT^{2}$$
53 $$1 + (58.3 + 58.3i)T + 2.80e3iT^{2}$$
59 $$1 + 13.7iT - 3.48e3T^{2}$$
61 $$1 - 39.0T + 3.72e3T^{2}$$
67 $$1 + (16.9 - 16.9i)T - 4.48e3iT^{2}$$
71 $$1 - 103.T + 5.04e3T^{2}$$
73 $$1 + (-41.5 - 41.5i)T + 5.32e3iT^{2}$$
79 $$1 - 155. iT - 6.24e3T^{2}$$
83 $$1 + (-12.7 - 12.7i)T + 6.88e3iT^{2}$$
89 $$1 + 141. iT - 7.92e3T^{2}$$
97 $$1 + (29.4 - 29.4i)T - 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$