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Properties

Label 2-567-63.16-c1-0-0
Degree $2$
Conductor $567$
Sign $0.0859 - 0.996i$
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  + (−0.730 − 1.26i)2-s + (−0.0665 + 0.115i)4-s + 0.593·5-s + (−2.25 + 1.38i)7-s − 2.72·8-s + (−0.433 − 0.750i)10-s − 4.46·11-s + (2.25 + 3.90i)13-s + (3.39 + 1.84i)14-s + (2.12 + 3.67i)16-s + (−0.136 − 0.236i)17-s + (−1.43 + 2.48i)19-s + (−0.0394 + 0.0684i)20-s + (3.25 + 5.64i)22-s − 5.05·23-s + ⋯
L(s)  = 1  + (−0.516 − 0.894i)2-s + (−0.0332 + 0.0576i)4-s + 0.265·5-s + (−0.853 + 0.521i)7-s − 0.964·8-s + (−0.137 − 0.237i)10-s − 1.34·11-s + (0.626 + 1.08i)13-s + (0.907 + 0.493i)14-s + (0.531 + 0.919i)16-s + (−0.0331 − 0.0574i)17-s + (−0.328 + 0.569i)19-s + (−0.00883 + 0.0152i)20-s + (0.694 + 1.20i)22-s − 1.05·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0859 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0859 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.0859 - 0.996i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.0859 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196157 + 0.179973i\)
\(L(\frac12)\) \(\approx\) \(0.196157 + 0.179973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 - 1.38i)T \)
good2 \( 1 + (0.730 + 1.26i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.593T + 5T^{2} \)
11 \( 1 + 4.46T + 11T^{2} \)
13 \( 1 + (-2.25 - 3.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.136 + 0.236i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.43 - 2.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 + (-0.176 + 0.305i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.32 + 5.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.44 + 9.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.69 - 2.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.21 - 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.66 - 9.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.02 - 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.36 - 2.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + (5.55 + 9.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.58 + 9.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.27 + 14.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.68 - 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.13 + 1.96i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68305545623257911755735704518, −10.23276304930113252945856518814, −9.318575248689314295322202401175, −8.726729285998870659498137437867, −7.51225226042373827613167020584, −6.15236446663943228423611473399, −5.71837334074950898061225077068, −4.01303251397645902024778208966, −2.76705405286937190546036829431, −1.86168441207883477619592440741, 0.16334105034755620271917900884, 2.62929070811697126180283207362, 3.68620485888192923634182545186, 5.34529555985822158834470813182, 6.13797255366989004591834015345, 6.98020038545369707483128712031, 7.970632050266914156642387581566, 8.410164719770837533811108244260, 9.745571976662414414203794602134, 10.18378020383572727139460479861

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