# Properties

 Label 2-570-57.56-c1-0-8 Degree $2$ Conductor $570$ Sign $0.927 + 0.374i$ 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 − 2-s + (−1 + 1.41i)3-s + 4-s + i·5-s + (1 − 1.41i)6-s + 0.585·7-s − 8-s + (−1.00 − 2.82i)9-s − i·10-s − 5.41i·11-s + (−1 + 1.41i)12-s − 2.24i·13-s − 0.585·14-s + (−1.41 − i)15-s + 16-s − 2.82i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.577 + 0.816i)3-s + 0.5·4-s + 0.447i·5-s + (0.408 − 0.577i)6-s + 0.221·7-s − 0.353·8-s + (−0.333 − 0.942i)9-s − 0.316i·10-s − 1.63i·11-s + (−0.288 + 0.408i)12-s − 0.621i·13-s − 0.156·14-s + (−0.365 − 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.927 + 0.374i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (341, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ 0.927 + 0.374i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.766397 - 0.148984i$$ $$L(\frac12)$$ $$\approx$$ $$0.766397 - 0.148984i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (1 - 1.41i)T$$
5 $$1 - iT$$
19 $$1 + (-4.24 + i)T$$
good7 $$1 - 0.585T + 7T^{2}$$
11 $$1 + 5.41iT - 11T^{2}$$
13 $$1 + 2.24iT - 13T^{2}$$
17 $$1 + 2.82iT - 17T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 + 5.07T + 29T^{2}$$
31 $$1 + 6.82iT - 31T^{2}$$
37 $$1 + 2.24iT - 37T^{2}$$
41 $$1 - 11.0T + 41T^{2}$$
43 $$1 - 6.58T + 43T^{2}$$
47 $$1 + 3.17iT - 47T^{2}$$
53 $$1 - 3.17T + 53T^{2}$$
59 $$1 - 12.8T + 59T^{2}$$
61 $$1 + 4.48T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 - 2.82T + 71T^{2}$$
73 $$1 - 2.48T + 73T^{2}$$
79 $$1 + 13.3iT - 79T^{2}$$
83 $$1 - 12.8iT - 83T^{2}$$
89 $$1 + 10.5T + 89T^{2}$$
97 $$1 + 6.58iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$