Properties

 Label 2-570-95.37-c1-0-13 Degree $2$ Conductor $570$ Sign $-0.978 + 0.204i$ 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 + (−0.253 − 2.22i)5-s + 1.00·6-s + (−2.47 − 2.47i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.75 + 1.39i)10-s + 2.74·11-s + (−0.707 + 0.707i)12-s + (1.20 + 1.20i)13-s + 3.50·14-s + (−1.39 + 1.75i)15-s − 1.00·16-s + (−4.87 − 4.87i)17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.113 − 0.993i)5-s + 0.408·6-s + (−0.935 − 0.935i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.553 + 0.440i)10-s + 0.827·11-s + (−0.204 + 0.204i)12-s + (0.333 + 0.333i)13-s + 0.935·14-s + (−0.359 + 0.451i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $-0.978 + 0.204i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ -0.978 + 0.204i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.0347170 - 0.335798i$$ $$L(\frac12)$$ $$\approx$$ $$0.0347170 - 0.335798i$$ $$L(\frac{3}{2})$$ 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 + (0.253 + 2.22i)T$$
19 $$1 + (1.94 - 3.90i)T$$
good7 $$1 + (2.47 + 2.47i)T + 7iT^{2}$$
11 $$1 - 2.74T + 11T^{2}$$
13 $$1 + (-1.20 - 1.20i)T + 13iT^{2}$$
17 $$1 + (4.87 + 4.87i)T + 17iT^{2}$$
23 $$1 + (-0.0321 + 0.0321i)T - 23iT^{2}$$
29 $$1 + 6.50T + 29T^{2}$$
31 $$1 - 6.50iT - 31T^{2}$$
37 $$1 + (4.58 - 4.58i)T - 37iT^{2}$$
41 $$1 + 5.96iT - 41T^{2}$$
43 $$1 + (-5.39 + 5.39i)T - 43iT^{2}$$
47 $$1 + (-3.66 - 3.66i)T + 47iT^{2}$$
53 $$1 + (8.97 + 8.97i)T + 53iT^{2}$$
59 $$1 + 4.42T + 59T^{2}$$
61 $$1 + 2.95T + 61T^{2}$$
67 $$1 + (7.00 - 7.00i)T - 67iT^{2}$$
71 $$1 + 5.56iT - 71T^{2}$$
73 $$1 + (2.19 - 2.19i)T - 73iT^{2}$$
79 $$1 - 0.225T + 79T^{2}$$
83 $$1 + (3.87 - 3.87i)T - 83iT^{2}$$
89 $$1 - 9.13T + 89T^{2}$$
97 $$1 + (-8.76 + 8.76i)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

−10.20194332283918183695827840716, −9.244916884726688845082416293918, −8.711946488219374594093827103545, −7.45893944108351912983940097941, −6.81116567051728899799482403202, −5.98240595059851871494970012418, −4.77860042346309242565187169350, −3.76378153345659601048576948569, −1.60803696079533543218176037815, −0.23468687304048084947566864387, 2.20155277323086302692439126398, 3.32465009904882975528501660196, 4.25863970468387340597851830335, 6.05272938760906316874465102481, 6.40149704841424157770650464755, 7.61764982393746041993674641686, 8.982238663800216063094836818911, 9.306122883350531058024011912942, 10.43388350257403678961707544781, 11.03764490537274373075150873412