Properties

 Label 2-570-57.56-c1-0-12 Degree $2$ Conductor $570$ Sign $0.914 - 0.404i$ 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 + (−0.828 + 1.52i)3-s + 4-s − i·5-s + (−0.828 + 1.52i)6-s + 4.83·7-s + 8-s + (−1.62 − 2.52i)9-s − i·10-s − 2i·11-s + (−0.828 + 1.52i)12-s + 1.04i·13-s + 4.83·14-s + (1.52 + 0.828i)15-s + 16-s + 0.140i·17-s + ⋯
 L(s)  = 1 + 0.707·2-s + (−0.478 + 0.878i)3-s + 0.5·4-s − 0.447i·5-s + (−0.338 + 0.620i)6-s + 1.82·7-s + 0.353·8-s + (−0.542 − 0.840i)9-s − 0.316i·10-s − 0.603i·11-s + (−0.239 + 0.439i)12-s + 0.288i·13-s + 1.29·14-s + (0.392 + 0.213i)15-s + 0.250·16-s + 0.0339i·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.914 - 0.404i$ 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: $$(2,\ 570,\ (\ :1/2),\ 0.914 - 0.404i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.21001 + 0.466691i$$ $$L(\frac12)$$ $$\approx$$ $$2.21001 + 0.466691i$$ $$L(\frac{3}{2})$$ 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 + (0.828 - 1.52i)T$$
5 $$1 + iT$$
19 $$1 + (-2.65 + 3.45i)T$$
good7 $$1 - 4.83T + 7T^{2}$$
11 $$1 + 2iT - 11T^{2}$$
13 $$1 - 1.04iT - 13T^{2}$$
17 $$1 - 0.140iT - 17T^{2}$$
23 $$1 - 6.63iT - 23T^{2}$$
29 $$1 + 1.65T + 29T^{2}$$
31 $$1 - 3.58iT - 31T^{2}$$
37 $$1 - 6.36iT - 37T^{2}$$
41 $$1 + 1.18T + 41T^{2}$$
43 $$1 + 8.76T + 43T^{2}$$
47 $$1 + 4.76iT - 47T^{2}$$
53 $$1 - 8.42T + 53T^{2}$$
59 $$1 - 0.350T + 59T^{2}$$
61 $$1 + 10.4T + 61T^{2}$$
67 $$1 + 13.6iT - 67T^{2}$$
71 $$1 + 11.7T + 71T^{2}$$
73 $$1 + 2.83T + 73T^{2}$$
79 $$1 - 10.0iT - 79T^{2}$$
83 $$1 + 10.0iT - 83T^{2}$$
89 $$1 + 16.4T + 89T^{2}$$
97 $$1 + 7.67iT - 97T^{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

−11.15648165677127295917132329510, −10.13241922869102860839000278490, −8.999743545250762201362075047290, −8.233428333138003747627912220837, −7.12326098786524890458460461463, −5.74085337634174894336338968406, −5.08520936425514045040351055959, −4.46373620669476606893037496390, −3.28907727686202117499889749047, −1.48543300797896228836544286738, 1.51477449168607220591880520593, 2.51734000833866675669940784817, 4.25752429350191604217897996941, 5.17623454458643915338880145215, 5.97321948829201935432346440706, 7.14824913347893468866812687173, 7.72625904813048555210817072646, 8.540979316603466810279749919367, 10.24701004944703738801201577232, 10.94938073169735128172872860810