# Properties

 Label 2-570-95.64-c1-0-1 Degree $2$ Conductor $570$ Sign $-0.939 + 0.342i$ Analytic cond. $4.55147$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.445 + 2.19i)5-s + (0.499 − 0.866i)6-s + 4.67i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.710 − 2.12i)10-s − 3.96·11-s + 0.999i·12-s + (0.698 + 0.403i)13-s + (−2.33 − 4.04i)14-s + (−0.710 − 2.12i)15-s + (−0.5 − 0.866i)16-s + (4.01 − 2.31i)17-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.199 + 0.979i)5-s + (0.204 − 0.353i)6-s + 1.76i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.224 − 0.670i)10-s − 1.19·11-s + 0.288i·12-s + (0.193 + 0.111i)13-s + (−0.624 − 1.08i)14-s + (−0.183 − 0.547i)15-s + (−0.125 − 0.216i)16-s + (0.974 − 0.562i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $-0.939 + 0.342i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (349, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ -0.939 + 0.342i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0873263 - 0.494181i$$ $$L(\frac12)$$ $$\approx$$ $$0.0873263 - 0.494181i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1 + (0.866 - 0.5i)T$$
5 $$1 + (0.445 - 2.19i)T$$
19 $$1 + (-3.01 - 3.15i)T$$
good7 $$1 - 4.67iT - 7T^{2}$$
11 $$1 + 3.96T + 11T^{2}$$
13 $$1 + (-0.698 - 0.403i)T + (6.5 + 11.2i)T^{2}$$
17 $$1 + (-4.01 + 2.31i)T + (8.5 - 14.7i)T^{2}$$
23 $$1 + (5.52 + 3.19i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (2.03 - 3.51i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 3.35T + 31T^{2}$$
37 $$1 + 2.19iT - 37T^{2}$$
41 $$1 + (2.02 + 3.51i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (6.36 - 3.67i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (8.12 + 4.69i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (1.34 + 0.778i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-2.20 + 3.81i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-6.42 - 3.70i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-1.08 - 1.87i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-10.2 + 5.91i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (4.67 + 8.09i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 9.92iT - 83T^{2}$$
89 $$1 + (9.13 - 15.8i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (10.7 - 6.20i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$