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Properties

Label 2-570-95.12-c1-0-4
Degree $2$
Conductor $570$
Sign $-0.309 - 0.950i$
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  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.347 + 2.20i)5-s + (−0.499 + 0.866i)6-s + (0.475 + 0.475i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.907 + 2.04i)10-s + 1.62·11-s + (−0.707 + 0.707i)12-s + (1.20 − 0.323i)13-s + (0.336 + 0.582i)14-s + (−2.04 − 0.907i)15-s + (0.500 + 0.866i)16-s + (−0.715 + 2.66i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.155 + 0.987i)5-s + (−0.204 + 0.353i)6-s + (0.179 + 0.179i)7-s + (0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.286 + 0.646i)10-s + 0.490·11-s + (−0.204 + 0.204i)12-s + (0.335 − 0.0898i)13-s + (0.0898 + 0.155i)14-s + (−0.527 − 0.234i)15-s + (0.125 + 0.216i)16-s + (−0.173 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16394 + 1.60306i\)
\(L(\frac12)\) \(\approx\) \(1.16394 + 1.60306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.347 - 2.20i)T \)
19 \( 1 + (3.74 - 2.23i)T \)
good7 \( 1 + (-0.475 - 0.475i)T + 7iT^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + (-1.20 + 0.323i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.715 - 2.66i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.346 - 1.29i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.85 - 4.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.96iT - 31T^{2} \)
37 \( 1 + (-1.34 + 1.34i)T - 37iT^{2} \)
41 \( 1 + (-3.11 + 1.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.02 - 1.07i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-10.0 + 2.70i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.10 + 0.297i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.21 - 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.415 - 0.719i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.23 + 4.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.55 + 3.78i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.5 - 3.10i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.50 + 6.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.66 + 4.66i)T - 83iT^{2} \)
89 \( 1 + (0.342 - 0.593i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.83 + 1.83i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.94865818656314492566086008437, −10.47129423457574672880536616667, −9.299012691405523028267065300451, −8.263413578667926310776477260863, −7.23697133482875966448689327591, −6.28089841275889640280920559622, −5.58869040934012783359056635525, −4.19437053802304465427009813156, −3.56672891444679940045382539344, −2.20784643431454045648063703823, 0.972985321430403935950718408023, 2.35599032599732809051324231663, 3.95290503109251654344941202858, 4.76984729639502370562866772535, 5.79928777142367191971108114291, 6.72709122146806913179347611107, 7.71925913466289417871706993531, 8.696149254518993366229126691023, 9.514993294096110117322202739929, 10.83484367907951058229168820817

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