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Properties

Label 2-567-189.185-c1-0-6
Degree $2$
Conductor $567$
Sign $0.986 - 0.166i$
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  + (−0.245 + 0.0432i)2-s + (−1.82 + 0.662i)4-s + (−1.99 + 0.727i)5-s + (0.302 − 2.62i)7-s + (0.848 − 0.489i)8-s + (0.458 − 0.264i)10-s + (0.489 − 1.34i)11-s + (1.30 + 3.59i)13-s + (0.0394 + 0.657i)14-s + (2.78 − 2.33i)16-s + (−0.109 − 0.188i)17-s + (2.56 + 1.48i)19-s + (3.15 − 2.64i)20-s + (−0.0618 + 0.350i)22-s + (7.61 + 1.34i)23-s + ⋯
L(s)  = 1  + (−0.173 + 0.0305i)2-s + (−0.910 + 0.331i)4-s + (−0.893 + 0.325i)5-s + (0.114 − 0.993i)7-s + (0.300 − 0.173i)8-s + (0.144 − 0.0836i)10-s + (0.147 − 0.405i)11-s + (0.362 + 0.996i)13-s + (0.0105 + 0.175i)14-s + (0.695 − 0.583i)16-s + (−0.0264 − 0.0458i)17-s + (0.589 + 0.340i)19-s + (0.706 − 0.592i)20-s + (−0.0131 + 0.0747i)22-s + (1.58 + 0.279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928833 + 0.0777746i\)
\(L(\frac12)\) \(\approx\) \(0.928833 + 0.0777746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.302 + 2.62i)T \)
good2 \( 1 + (0.245 - 0.0432i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (1.99 - 0.727i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.489 + 1.34i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.30 - 3.59i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.109 + 0.188i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.56 - 1.48i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.61 - 1.34i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.78 + 4.90i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.32 - 3.64i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 - 6.01T + 37T^{2} \)
41 \( 1 + (-10.0 + 3.66i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.29 - 7.34i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-5.51 - 2.00i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (12.1 + 6.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.21 - 1.01i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.45 + 3.99i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.59 + 14.6i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.29 - 2.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.46iT - 73T^{2} \)
79 \( 1 + (0.675 + 3.83i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.141 - 0.0515i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (3.40 - 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.2 + 2.16i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.96234716242085435756803277364, −9.722603695810839294365955024427, −9.058173095394542983315550123630, −7.948639289230419239892213259069, −7.48257936196678968553167925506, −6.40650058792200358005130598346, −4.86182758766691744262923992313, −4.05404909443438421619997242983, −3.27988260168866328604114831341, −0.926525440078497777496477245194, 0.908709024676711738064963953329, 2.88376508974279906581282960713, 4.20858372776923001713699343250, 5.07548684956124294464957713703, 5.91870202432132486304781188586, 7.39633765579042512228490608211, 8.263289936005146293221021858727, 8.920919815986273985544348691795, 9.643861073081239999284218481017, 10.75722116642655275083415386795

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