Properties

Label 2-570-285.29-c1-0-22
Degree $2$
Conductor $570$
Sign $0.463 + 0.886i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 + 0.173i)2-s + (1.43 − 0.974i)3-s + (0.939 − 0.342i)4-s + (−2.18 + 0.487i)5-s + (−1.24 + 1.20i)6-s + (1.28 + 0.739i)7-s + (−0.866 + 0.5i)8-s + (1.10 − 2.79i)9-s + (2.06 − 0.858i)10-s + (2.81 − 1.62i)11-s + (1.01 − 1.40i)12-s + (−3.29 − 2.76i)13-s + (−1.38 − 0.505i)14-s + (−2.65 + 2.82i)15-s + (0.766 − 0.642i)16-s + (−0.748 − 4.24i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (0.826 − 0.562i)3-s + (0.469 − 0.171i)4-s + (−0.975 + 0.217i)5-s + (−0.506 + 0.493i)6-s + (0.483 + 0.279i)7-s + (−0.306 + 0.176i)8-s + (0.367 − 0.930i)9-s + (0.652 − 0.271i)10-s + (0.849 − 0.490i)11-s + (0.292 − 0.405i)12-s + (−0.913 − 0.766i)13-s + (−0.371 − 0.135i)14-s + (−0.684 + 0.729i)15-s + (0.191 − 0.160i)16-s + (−0.181 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.463 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04830 - 0.634755i\)
\(L(\frac12)\) \(\approx\) \(1.04830 - 0.634755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-1.43 + 0.974i)T \)
5 \( 1 + (2.18 - 0.487i)T \)
19 \( 1 + (-1.45 - 4.10i)T \)
good7 \( 1 + (-1.28 - 0.739i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.81 + 1.62i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.29 + 2.76i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.748 + 4.24i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-7.23 + 2.63i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.791 + 4.48i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.19 - 1.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.11T + 37T^{2} \)
41 \( 1 + (1.02 - 0.862i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.81 + 7.73i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.06 - 11.6i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.503 - 1.38i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.10 + 11.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.53 + 3.10i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.31 - 13.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.07 + 0.757i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.62 + 9.08i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-4.55 - 5.43i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.29 + 2.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.44 - 2.89i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.63 - 9.26i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.50369137109623990746178496325, −9.443555361693185199277800813892, −8.681902620764936430722897555146, −7.945182745715962005347133793762, −7.30250345327868637280644755881, −6.46809428379642629962689148997, −4.95857807957601950129023357039, −3.51636350166421579602699238504, −2.55169406460618389115684714687, −0.876996959737437239337859847867, 1.57268642305142127687365590778, 3.06782917925285101482685128408, 4.19379106347386647920101537624, 4.91590158980551132700589915291, 6.97731407237178480611091819655, 7.38221661729083900053691561511, 8.509701723730672988472528039081, 9.006838915445444670061976313938, 9.835697762835462179203772810358, 10.86455544981866293937818525384

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