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Properties

Label 2-567-7.2-c1-0-18
Degree $2$
Conductor $567$
Sign $0.925 + 0.377i$
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  + (−0.635 + 1.10i)2-s + (0.193 + 0.334i)4-s + (0.776 − 1.34i)5-s + (−1.48 − 2.18i)7-s − 3.03·8-s + (0.986 + 1.70i)10-s + (−1.60 − 2.77i)11-s + 4.78·13-s + (3.35 − 0.245i)14-s + (1.53 − 2.66i)16-s + (−1.05 − 1.83i)17-s + (2.43 − 4.21i)19-s + 0.600·20-s + 4.07·22-s + (1.85 − 3.21i)23-s + ⋯
L(s)  = 1  + (−0.449 + 0.777i)2-s + (0.0966 + 0.167i)4-s + (0.347 − 0.601i)5-s + (−0.561 − 0.827i)7-s − 1.07·8-s + (0.311 + 0.540i)10-s + (−0.483 − 0.838i)11-s + 1.32·13-s + (0.895 − 0.0655i)14-s + (0.384 − 0.666i)16-s + (−0.256 − 0.444i)17-s + (0.557 − 0.966i)19-s + 0.134·20-s + 0.869·22-s + (0.386 − 0.669i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.925 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02994 - 0.201900i\)
\(L(\frac12)\) \(\approx\) \(1.02994 - 0.201900i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 2.18i)T \)
good2 \( 1 + (0.635 - 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.776 + 1.34i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.60 + 2.77i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.78T + 13T^{2} \)
17 \( 1 + (1.05 + 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.43 + 4.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 3.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.37T + 29T^{2} \)
31 \( 1 + (2.75 + 4.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0932 + 0.161i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (-0.885 + 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.834 - 1.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.91 - 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 - 5.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.11 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (5.93 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.654 + 1.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.346T + 83T^{2} \)
89 \( 1 + (8.70 - 15.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.81769901025726592951364884892, −9.252901163398461859266344335800, −9.092300668377523490104933086289, −7.933283396821627961407420748195, −7.19946085815776074033992042766, −6.23528190799873408384716248044, −5.46855281463627861861516463701, −3.96586057341497644931323163678, −2.87320330182179210669973616218, −0.70758864984472242576688756720, 1.62563614773303150125020108312, 2.67201319597907113378013633941, 3.69631517297824351070651649362, 5.56484173657085013348049975795, 6.09163611903928612930408125871, 7.15672425014332638194147157787, 8.460168848099577410334838224152, 9.348905945223813436282449404450, 9.943233139257572878594777427277, 10.81238265963602489240057257342

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