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Properties

Label 2-570-57.56-c3-0-0
Degree $2$
Conductor $570$
Sign $-0.957 + 0.287i$
Analytic cond. $33.6310$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (−5.18 + 0.264i)3-s + 4·4-s + 5i·5-s + (−10.3 + 0.529i)6-s − 11.4·7-s + 8·8-s + (26.8 − 2.74i)9-s + 10i·10-s − 20.9i·11-s + (−20.7 + 1.05i)12-s + 3.31i·13-s − 22.8·14-s + (−1.32 − 25.9i)15-s + 16·16-s + 76.3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.998 + 0.0509i)3-s + 0.5·4-s + 0.447i·5-s + (−0.706 + 0.0360i)6-s − 0.616·7-s + 0.353·8-s + (0.994 − 0.101i)9-s + 0.316i·10-s − 0.573i·11-s + (−0.499 + 0.0254i)12-s + 0.0707i·13-s − 0.436·14-s + (−0.0227 − 0.446i)15-s + 0.250·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.957 + 0.287i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ -0.957 + 0.287i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.002819757864\)
\(L(\frac12)\) \(\approx\) \(0.002819757864\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (5.18 - 0.264i)T \)
5 \( 1 - 5iT \)
19 \( 1 + (-19.7 - 80.4i)T \)
good7 \( 1 + 11.4T + 343T^{2} \)
11 \( 1 + 20.9iT - 1.33e3T^{2} \)
13 \( 1 - 3.31iT - 2.19e3T^{2} \)
17 \( 1 - 76.3iT - 4.91e3T^{2} \)
23 \( 1 + 127. iT - 1.21e4T^{2} \)
29 \( 1 + 250.T + 2.43e4T^{2} \)
31 \( 1 - 116. iT - 2.97e4T^{2} \)
37 \( 1 + 412. iT - 5.06e4T^{2} \)
41 \( 1 + 414.T + 6.89e4T^{2} \)
43 \( 1 - 368.T + 7.95e4T^{2} \)
47 \( 1 + 258. iT - 1.03e5T^{2} \)
53 \( 1 + 475.T + 1.48e5T^{2} \)
59 \( 1 - 99.2T + 2.05e5T^{2} \)
61 \( 1 + 757.T + 2.26e5T^{2} \)
67 \( 1 + 289. iT - 3.00e5T^{2} \)
71 \( 1 + 796.T + 3.57e5T^{2} \)
73 \( 1 + 219.T + 3.89e5T^{2} \)
79 \( 1 - 485. iT - 4.93e5T^{2} \)
83 \( 1 + 91.7iT - 5.71e5T^{2} \)
89 \( 1 + 299.T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3iT - 9.12e5T^{2} \)
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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

−10.74628212241089191808941802930, −10.41474528256517623740581266019, −9.239315214275799243360798529481, −7.87938634397934627418791057081, −6.86576366223755357625420369843, −6.10379866792692188364715723135, −5.50628557880174817264910075507, −4.17243836789650359470928521369, −3.36338675791431409306271385968, −1.75070820724372367068128189297, 0.00072768198919704891023860262, 1.51392458051852579590011358638, 3.10258314338304534961550081874, 4.41923977898231130125988807203, 5.14299332627809110085767574292, 6.01458722195312228526843388003, 6.98582182076123113739224452507, 7.65203566526115707433988177771, 9.336575294754374963481994939353, 9.833896834904170647374521294781

Graph of the $Z$-function along the critical line