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Properties

Label 2-570-19.11-c3-0-3
Degree $2$
Conductor $570$
Sign $0.388 - 0.921i$
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  + (1 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (2.5 − 4.33i)5-s + (−3 − 5.19i)6-s − 9.46·7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−5 − 8.66i)10-s − 31.3·11-s − 12·12-s + (32.7 + 56.6i)13-s + (−9.46 + 16.3i)14-s + (−7.50 − 12.9i)15-s + (−8 + 13.8i)16-s + (−56.3 + 97.5i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.204 − 0.353i)6-s − 0.511·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s − 0.860·11-s − 0.288·12-s + (0.698 + 1.20i)13-s + (−0.180 + 0.312i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.803 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6585729119\)
\(L(\frac12)\) \(\approx\) \(0.6585729119\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
19 \( 1 + (82.4 + 7.33i)T \)
good7 \( 1 + 9.46T + 343T^{2} \)
11 \( 1 + 31.3T + 1.33e3T^{2} \)
13 \( 1 + (-32.7 - 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (56.3 - 97.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-25.4 - 44.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (60.3 + 104. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 209.T + 2.97e4T^{2} \)
37 \( 1 + 164.T + 5.06e4T^{2} \)
41 \( 1 + (72.6 - 125. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (7.66 - 13.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-261. - 453. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (146. + 253. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (2.90 - 5.02i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (356. + 617. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-27.7 - 48.0i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (112. - 194. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (177. - 307. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (261. - 453. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 499.T + 5.71e5T^{2} \)
89 \( 1 + (-148. - 257. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (182. - 315. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.62213873256707550876989434883, −9.630913162443306553309251099904, −8.785473652207676100931674522036, −8.058886623666784647167875806097, −6.59066030904246029191251555719, −6.07168185043306979992381906608, −4.67760183437504803775880343977, −3.75265531057865727235587906633, −2.43375007685361610707691196924, −1.50083571953796585349838885039, 0.15727179410324322326804221295, 2.57068449770902692292624264936, 3.35607680575835480389713233441, 4.63516778648210098947533727171, 5.52850133254690730597333454133, 6.49100134590152023275551099610, 7.41159976811041046321296304073, 8.419185847330040276237044146410, 9.127366855109508212752975086083, 10.31397728823568082436203195068

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