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Properties

Label 2-570-285.179-c1-0-14
Degree $2$
Conductor $570$
Sign $0.231 - 0.972i$
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  + (−0.866 − 0.5i)2-s + (1.08 + 1.34i)3-s + (0.499 + 0.866i)4-s + (2.16 + 0.554i)5-s + (−0.267 − 1.71i)6-s + 1.36i·7-s − 0.999i·8-s + (−0.635 + 2.93i)9-s + (−1.59 − 1.56i)10-s + 5.06i·11-s + (−0.623 + 1.61i)12-s + (−1.22 − 2.11i)13-s + (0.681 − 1.18i)14-s + (1.60 + 3.52i)15-s + (−0.5 + 0.866i)16-s + (−1.98 + 3.43i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.627 + 0.778i)3-s + (0.249 + 0.433i)4-s + (0.968 + 0.247i)5-s + (−0.109 − 0.698i)6-s + 0.515i·7-s − 0.353i·8-s + (−0.211 + 0.977i)9-s + (−0.505 − 0.494i)10-s + 1.52i·11-s + (−0.180 + 0.466i)12-s + (−0.338 − 0.586i)13-s + (0.182 − 0.315i)14-s + (0.415 + 0.909i)15-s + (−0.125 + 0.216i)16-s + (−0.481 + 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15667 + 0.913365i\)
\(L(\frac12)\) \(\approx\) \(1.15667 + 0.913365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.08 - 1.34i)T \)
5 \( 1 + (-2.16 - 0.554i)T \)
19 \( 1 + (-1.02 + 4.23i)T \)
good7 \( 1 - 1.36iT - 7T^{2} \)
11 \( 1 - 5.06iT - 11T^{2} \)
13 \( 1 + (1.22 + 2.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.98 - 3.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.36 - 4.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.32iT - 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 + (2.09 - 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.08 + 4.66i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.62 - 8.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.37 + 3.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.294 - 0.510i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.56 - 6.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.50 - 9.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.35 + 9.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.55 - 3.78i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.44 + 3.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (3.62 + 6.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.42 + 9.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.41293819764945215782407976268, −10.11219781255169652458493000304, −9.288629590497910862105697886806, −8.616799874457332306110566569057, −7.57200938217886653838204691310, −6.50290137236681457636290702255, −5.22942089098311083689281759283, −4.23035835392032558848084116424, −2.67532413170048672753601939542, −2.09681411159514389416536690147, 0.979425241446810424211475453948, 2.21171322302991744978427371834, 3.55336029328846191758025647958, 5.31365906738826131114934602244, 6.24601952621388226651728236656, 6.97937594784015539472535217288, 8.008430471687987449357991572117, 8.730574970274130861147473032872, 9.492738232613065565337430166536, 10.25846774377257402069245562179

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