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Properties

Label 2-567-189.185-c1-0-5
Degree $2$
Conductor $567$
Sign $0.611 - 0.791i$
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  + (−2.22 + 0.391i)2-s + (2.90 − 1.05i)4-s + (3.20 − 1.16i)5-s + (−2.62 − 0.334i)7-s + (−2.12 + 1.22i)8-s + (−6.66 + 3.84i)10-s + (−0.870 + 2.39i)11-s + (2.11 + 5.80i)13-s + (5.96 − 0.285i)14-s + (−0.494 + 0.414i)16-s + (1.19 + 2.07i)17-s + (−1.07 − 0.622i)19-s + (8.07 − 6.77i)20-s + (0.996 − 5.65i)22-s + (2.21 + 0.390i)23-s + ⋯
L(s)  = 1  + (−1.57 + 0.276i)2-s + (1.45 − 0.527i)4-s + (1.43 − 0.522i)5-s + (−0.991 − 0.126i)7-s + (−0.750 + 0.433i)8-s + (−2.10 + 1.21i)10-s + (−0.262 + 0.720i)11-s + (0.585 + 1.60i)13-s + (1.59 − 0.0763i)14-s + (−0.123 + 0.103i)16-s + (0.290 + 0.503i)17-s + (−0.247 − 0.142i)19-s + (1.80 − 1.51i)20-s + (0.212 − 1.20i)22-s + (0.461 + 0.0814i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.611 - 0.791i$
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: \((2,\ 567,\ (\ :1/2),\ 0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682088 + 0.335168i\)
\(L(\frac12)\) \(\approx\) \(0.682088 + 0.335168i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.334i)T \)
good2 \( 1 + (2.22 - 0.391i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (-3.20 + 1.16i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (0.870 - 2.39i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.11 - 5.80i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.19 - 2.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.07 + 0.622i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 0.390i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.685 - 1.88i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.538 + 1.48i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + (5.39 - 1.96i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.110 + 0.625i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-11.3 - 4.12i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.15 - 2.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.11 + 1.77i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.52 + 6.92i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.39 - 7.89i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.45 + 2.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + (-0.167 - 0.951i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.38 + 0.867i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-4.97 + 8.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.93 - 1.39i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.36972733465946710584905625801, −9.729741017567060132998202006849, −9.238137682421067475068823464041, −8.617094424550442035236839630943, −7.29694307777810632907265972265, −6.54161015642972015406829354556, −5.82785987580708393400997037743, −4.30230607943530485523978563504, −2.34368692343454563119863841266, −1.32081691120381914627224370625, 0.804355030833810951639166452750, 2.43329656970798116762873501313, 3.17959836160513391561315584081, 5.53342639530191947203830919886, 6.17940428144576214260156666041, 7.19284524797226714234138228605, 8.220731027550599675926467335450, 9.097417675372530756733244992289, 9.757681935110647209633632937882, 10.48223023456407392391001818091

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