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Properties

Label 2-567-189.101-c1-0-14
Degree $2$
Conductor $567$
Sign $0.783 + 0.621i$
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  + (0.582 − 0.694i)2-s + (0.204 + 1.16i)4-s + (0.931 − 0.781i)5-s + (−2.06 − 1.65i)7-s + (2.49 + 1.44i)8-s − 1.10i·10-s + (2.30 − 2.74i)11-s + (0.206 − 0.566i)13-s + (−2.35 + 0.473i)14-s + (0.238 − 0.0867i)16-s + 7.94·17-s − 1.41i·19-s + (1.09 + 0.920i)20-s + (−0.564 − 3.20i)22-s + (0.349 − 0.958i)23-s + ⋯
L(s)  = 1  + (0.411 − 0.490i)2-s + (0.102 + 0.580i)4-s + (0.416 − 0.349i)5-s + (−0.781 − 0.623i)7-s + (0.882 + 0.509i)8-s − 0.348i·10-s + (0.695 − 0.828i)11-s + (0.0571 − 0.157i)13-s + (−0.628 + 0.126i)14-s + (0.0595 − 0.0216i)16-s + 1.92·17-s − 0.324i·19-s + (0.245 + 0.205i)20-s + (−0.120 − 0.682i)22-s + (0.0727 − 0.199i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91993 - 0.668440i\)
\(L(\frac12)\) \(\approx\) \(1.91993 - 0.668440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 + 1.65i)T \)
good2 \( 1 + (-0.582 + 0.694i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.931 + 0.781i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.30 + 2.74i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.206 + 0.566i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 7.94T + 17T^{2} \)
19 \( 1 + 1.41iT - 19T^{2} \)
23 \( 1 + (-0.349 + 0.958i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.413 - 1.13i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-8.17 + 1.44i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.22 - 7.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.73 + 2.45i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.41 + 8.02i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.0548 + 0.310i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (5.74 + 3.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.936 - 0.341i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (12.0 + 2.12i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.90 - 6.63i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.952 + 0.550i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.808 - 0.466i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.3 + 8.64i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-5.66 + 2.06i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 5.92T + 89T^{2} \)
97 \( 1 + (8.29 + 1.46i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.63032236234829942841792408946, −9.969614804546301312685447650331, −8.926397666158142622293340871503, −8.009267117543802926618112513838, −7.05048449216341802202335043279, −6.01830617963822912546396539146, −4.89826002595260873484018738083, −3.62728560288145586612437955942, −3.07351267801102962344553408364, −1.27312288813674820953279516346, 1.55131551150344131214775577489, 3.04198247888626606260851111219, 4.38749135356261039222371098580, 5.54874122214049483934098902412, 6.21803092774801377726455409667, 6.92791322901264602552140425723, 7.993642759956383895498595710772, 9.427994896700117989575314897949, 9.862653280491533727502850258949, 10.58288683653401171429964937558

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