# Properties

 Label 2-570-95.18-c1-0-8 Degree $2$ Conductor $570$ Sign $0.555 + 0.831i$ 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 + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−2.23 − 0.0685i)5-s + 1.00·6-s + (−2.16 + 2.16i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (1.53 + 1.62i)10-s − 5.68·11-s + (−0.707 − 0.707i)12-s + (3.92 − 3.92i)13-s + 3.06·14-s + (1.62 − 1.53i)15-s − 1.00·16-s + (4.99 − 4.99i)17-s + ⋯
 L(s)  = 1 + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.999 − 0.0306i)5-s + 0.408·6-s + (−0.818 + 0.818i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.484 + 0.515i)10-s − 1.71·11-s + (−0.204 − 0.204i)12-s + (1.08 − 1.08i)13-s + 0.818·14-s + (0.420 − 0.395i)15-s − 0.250·16-s + (1.21 − 1.21i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.555 + 0.831i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (493, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ 0.555 + 0.831i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.510852 - 0.273024i$$ $$L(\frac12)$$ $$\approx$$ $$0.510852 - 0.273024i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 + 0.707i)T$$
3 $$1 + (0.707 - 0.707i)T$$
5 $$1 + (2.23 + 0.0685i)T$$
19 $$1 + (-4.01 - 1.68i)T$$
good7 $$1 + (2.16 - 2.16i)T - 7iT^{2}$$
11 $$1 + 5.68T + 11T^{2}$$
13 $$1 + (-3.92 + 3.92i)T - 13iT^{2}$$
17 $$1 + (-4.99 + 4.99i)T - 17iT^{2}$$
23 $$1 + (-4.30 - 4.30i)T + 23iT^{2}$$
29 $$1 + 4.04T + 29T^{2}$$
31 $$1 + 4.04iT - 31T^{2}$$
37 $$1 + (-6.19 - 6.19i)T + 37iT^{2}$$
41 $$1 + 6.38iT - 41T^{2}$$
43 $$1 + (4.15 + 4.15i)T + 43iT^{2}$$
47 $$1 + (-3.24 + 3.24i)T - 47iT^{2}$$
53 $$1 + (-5.40 + 5.40i)T - 53iT^{2}$$
59 $$1 + 2.39T + 59T^{2}$$
61 $$1 + 2.33T + 61T^{2}$$
67 $$1 + (6.12 + 6.12i)T + 67iT^{2}$$
71 $$1 + 6.51iT - 71T^{2}$$
73 $$1 + (2.07 + 2.07i)T + 73iT^{2}$$
79 $$1 - 4.23T + 79T^{2}$$
83 $$1 + (5.30 + 5.30i)T + 83iT^{2}$$
89 $$1 + 2.50T + 89T^{2}$$
97 $$1 + (-6.22 - 6.22i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$