# Properties

 Label 2-570-15.2-c1-0-4 Degree $2$ Conductor $570$ Sign $0.304 - 0.952i$ Analytic cond. $4.55147$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)2-s + (−1.35 − 1.07i)3-s − 1.00i·4-s + (1.25 + 1.85i)5-s + (1.72 − 0.194i)6-s + (−3.56 − 3.56i)7-s + (0.707 + 0.707i)8-s + (0.668 + 2.92i)9-s + (−2.19 − 0.421i)10-s + 3.35i·11-s + (−1.07 + 1.35i)12-s + (0.759 − 0.759i)13-s + 5.03·14-s + (0.298 − 3.86i)15-s − 1.00·16-s + (0.534 − 0.534i)17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + (−0.781 − 0.623i)3-s − 0.500i·4-s + (0.561 + 0.827i)5-s + (0.702 − 0.0793i)6-s + (−1.34 − 1.34i)7-s + (0.250 + 0.250i)8-s + (0.222 + 0.974i)9-s + (−0.694 − 0.133i)10-s + 1.01i·11-s + (−0.311 + 0.390i)12-s + (0.210 − 0.210i)13-s + 1.34·14-s + (0.0771 − 0.997i)15-s − 0.250·16-s + (0.129 − 0.129i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.304 - 0.952i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (77, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ 0.304 - 0.952i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.555917 + 0.406058i$$ $$L(\frac12)$$ $$\approx$$ $$0.555917 + 0.406058i$$ $$L(\frac{3}{2})$$ 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 + (1.35 + 1.07i)T$$
5 $$1 + (-1.25 - 1.85i)T$$
19 $$1 - iT$$
good7 $$1 + (3.56 + 3.56i)T + 7iT^{2}$$
11 $$1 - 3.35iT - 11T^{2}$$
13 $$1 + (-0.759 + 0.759i)T - 13iT^{2}$$
17 $$1 + (-0.534 + 0.534i)T - 17iT^{2}$$
23 $$1 + (-3.95 - 3.95i)T + 23iT^{2}$$
29 $$1 + 3.01T + 29T^{2}$$
31 $$1 - 8.18T + 31T^{2}$$
37 $$1 + (-5.57 - 5.57i)T + 37iT^{2}$$
41 $$1 - 8.43iT - 41T^{2}$$
43 $$1 + (-4.29 + 4.29i)T - 43iT^{2}$$
47 $$1 + (8.00 - 8.00i)T - 47iT^{2}$$
53 $$1 + (-7.04 - 7.04i)T + 53iT^{2}$$
59 $$1 + 4.32T + 59T^{2}$$
61 $$1 - 1.35T + 61T^{2}$$
67 $$1 + (-5.23 - 5.23i)T + 67iT^{2}$$
71 $$1 + 4.74iT - 71T^{2}$$
73 $$1 + (5.12 - 5.12i)T - 73iT^{2}$$
79 $$1 - 0.256iT - 79T^{2}$$
83 $$1 + (4.73 + 4.73i)T + 83iT^{2}$$
89 $$1 - 15.2T + 89T^{2}$$
97 $$1 + (9.73 + 9.73i)T + 97iT^{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

−10.67849878147317983335677037867, −10.00053458076162407294203808252, −9.554352336170835421780583374012, −7.78958595567215396253456945996, −7.16471654292689392276478817966, −6.56684700223375882036554544886, −5.87800487912117647048127687442, −4.49598610519181413939487946264, −2.93903193352183222355418823864, −1.21571805911568411936068021391, 0.58496052096013581795242555685, 2.54767760578133885948197386434, 3.71145201177815288994613079223, 5.09964592222487442890362313547, 5.95008613303081190346055999433, 6.59011513114950582767632373732, 8.470688605381039959705648590850, 9.051755801483630821852082851597, 9.615040972317695353504966258365, 10.43722378680200132800272976478