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Properties

Label 2-567-63.4-c1-0-21
Degree $2$
Conductor $567$
Sign $0.474 + 0.880i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  + (−1.29 + 2.24i)2-s + (−2.34 − 4.06i)4-s + 2.28·5-s + (−2.08 − 1.63i)7-s + 6.97·8-s + (−2.95 + 5.11i)10-s − 2.95·11-s + (−2.13 + 3.69i)13-s + (6.34 − 2.56i)14-s + (−4.32 + 7.49i)16-s + (0.764 − 1.32i)17-s + (−3.69 − 6.39i)19-s + (−5.35 − 9.28i)20-s + (3.82 − 6.62i)22-s − 6.15·23-s + ⋯
L(s)  = 1  + (−0.914 + 1.58i)2-s + (−1.17 − 2.03i)4-s + 1.02·5-s + (−0.787 − 0.616i)7-s + 2.46·8-s + (−0.934 + 1.61i)10-s − 0.891·11-s + (−0.591 + 1.02i)13-s + (1.69 − 0.684i)14-s + (−1.08 + 1.87i)16-s + (0.185 − 0.321i)17-s + (−0.846 − 1.46i)19-s + (−1.19 − 2.07i)20-s + (0.815 − 1.41i)22-s − 1.28·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206908 - 0.123509i\)
\(L(\frac12)\) \(\approx\) \(0.206908 - 0.123509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.08 + 1.63i)T \)
good2 \( 1 + (1.29 - 2.24i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + (2.13 - 3.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.764 + 1.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
29 \( 1 + (-1.17 - 2.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.11 + 5.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.58 + 6.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.94 + 6.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.417 + 0.722i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.71 - 6.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.31 - 4.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.56 + 6.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.160T + 71T^{2} \)
73 \( 1 + (-0.190 + 0.329i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.97 + 6.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.14 + 3.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.661 - 1.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.13157261686031503306013795956, −9.486778985976792623575299908520, −8.949342026809977178739443277229, −7.70864135860045406087514511902, −7.02382245588953677847017341399, −6.26791318496112858642434691737, −5.47141299191055090886011923026, −4.37605729086767905638561359002, −2.22175572953440850901330972208, −0.17216227046461902905540869565, 1.81987741576025857086640531223, 2.66073305498472674897598189038, 3.65695191168223132791147927480, 5.28393469457247548529574255192, 6.26289742640017041865197920180, 7.930123131962110496063968360083, 8.448051118797600015026515261813, 9.622748764108280806936781684594, 10.14012119807402988016608843101, 10.40705756575758765177965035743

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