# Properties

 Label 2-570-1.1-c1-0-9 Degree $2$ Conductor $570$ Sign $-1$ Analytic cond. $4.55147$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 6·11-s − 12-s − 2·14-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 2·21-s + 6·22-s + 4·23-s + 24-s + 25-s − 27-s + 2·28-s − 8·29-s − 30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s + 1.27·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.182·30-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $-1$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{570} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + T$$
5 $$1 + T$$
19 $$1 + T$$
good7 $$1 - 2 T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 + 8 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 + 4 T + p T^{2}$$
43 $$1 + 6 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 + 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 + 4 T + p T^{2}$$
97 $$1 - 12 T + p T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.52962093414480357476507293182, −9.493939244992219316261816728318, −8.350027949722488694655969438693, −7.72731251340449738967673960420, −6.96220474060789094629124128282, −5.54119641438834227046404630981, −4.94104499551584536924889209574, −3.34597339745375604130568876294, −1.83499646563289112140885589168, 0, 1.83499646563289112140885589168, 3.34597339745375604130568876294, 4.94104499551584536924889209574, 5.54119641438834227046404630981, 6.96220474060789094629124128282, 7.72731251340449738967673960420, 8.350027949722488694655969438693, 9.493939244992219316261816728318, 10.52962093414480357476507293182