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Properties

Label 2-567-189.131-c1-0-1
Degree $2$
Conductor $567$
Sign $0.714 - 0.699i$
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  + (−1.61 − 1.92i)2-s + (−0.750 + 4.25i)4-s + (2.10 + 1.76i)5-s + (−0.122 + 2.64i)7-s + (5.05 − 2.91i)8-s − 6.90i·10-s + (−2.69 − 3.20i)11-s + (−0.0398 − 0.109i)13-s + (5.28 − 4.03i)14-s + (−5.65 − 2.05i)16-s − 5.14·17-s + 5.21i·19-s + (−9.08 + 7.62i)20-s + (−1.82 + 10.3i)22-s + (2.56 + 7.05i)23-s + ⋯
L(s)  = 1  + (−1.14 − 1.36i)2-s + (−0.375 + 2.12i)4-s + (0.940 + 0.789i)5-s + (−0.0463 + 0.998i)7-s + (1.78 − 1.03i)8-s − 2.18i·10-s + (−0.811 − 0.966i)11-s + (−0.0110 − 0.0303i)13-s + (1.41 − 1.07i)14-s + (−1.41 − 0.514i)16-s − 1.24·17-s + 1.19i·19-s + (−2.03 + 1.70i)20-s + (−0.389 + 2.20i)22-s + (0.535 + 1.47i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571229 + 0.233046i\)
\(L(\frac12)\) \(\approx\) \(0.571229 + 0.233046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.122 - 2.64i)T \)
good2 \( 1 + (1.61 + 1.92i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-2.10 - 1.76i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (2.69 + 3.20i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.0398 + 0.109i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 5.14T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 + (-2.56 - 7.05i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.21 - 3.34i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.305 + 0.0538i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.959 + 1.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.29 - 2.29i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.411 - 2.33i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.876 - 4.97i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.76 + 2.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.84 - 3.58i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.13 - 0.200i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.49 - 7.12i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-5.47 - 3.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.23 + 1.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.0 + 10.1i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-10.6 - 3.88i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 0.913T + 89T^{2} \)
97 \( 1 + (5.88 - 1.03i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.82957524265987420661770643589, −10.03609802523801953021536989646, −9.290344666162193352679072177775, −8.598029795813019020763954759796, −7.69168602183561438597343501749, −6.33978920402629406338488058099, −5.37918218961160361411531955317, −3.45894605611566005828846148782, −2.64785296890079501175331534190, −1.74609253651154536065624750889, 0.49492086410289288002077153006, 2.07384874712363060091004866803, 4.61945341859904988640501438951, 5.17797484619985087562191473822, 6.51322183358986867286471028711, 6.98624293760391177603318816774, 7.991827281700753862561688142300, 8.878623912042212550827119067784, 9.490810514611783242567052488643, 10.29124293201488338032017387933

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