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Properties

Label 2-567-189.101-c1-0-4
Degree $2$
Conductor $567$
Sign $-0.945 - 0.325i$
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  + (−1.47 + 1.76i)2-s + (−0.573 − 3.24i)4-s + (1.41 − 1.18i)5-s + (−2.40 + 1.11i)7-s + (2.59 + 1.49i)8-s + 4.24i·10-s + (0.0789 − 0.0940i)11-s + (0.0159 − 0.0437i)13-s + (1.58 − 5.87i)14-s + (−0.271 + 0.0988i)16-s − 0.157·17-s + 7.59i·19-s + (−4.65 − 3.90i)20-s + (0.0490 + 0.278i)22-s + (−0.332 + 0.913i)23-s + ⋯
L(s)  = 1  + (−1.04 + 1.24i)2-s + (−0.286 − 1.62i)4-s + (0.631 − 0.529i)5-s + (−0.907 + 0.420i)7-s + (0.916 + 0.528i)8-s + 1.34i·10-s + (0.0238 − 0.0283i)11-s + (0.00441 − 0.0121i)13-s + (0.424 − 1.57i)14-s + (−0.0678 + 0.0247i)16-s − 0.0381·17-s + 1.74i·19-s + (−1.04 − 0.874i)20-s + (0.0104 + 0.0593i)22-s + (−0.0693 + 0.190i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.945 - 0.325i$
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: \((2,\ 567,\ (\ :1/2),\ -0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0978531 + 0.585561i\)
\(L(\frac12)\) \(\approx\) \(0.0978531 + 0.585561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.40 - 1.11i)T \)
good2 \( 1 + (1.47 - 1.76i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.41 + 1.18i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.0789 + 0.0940i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.0159 + 0.0437i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.157T + 17T^{2} \)
19 \( 1 - 7.59iT - 19T^{2} \)
23 \( 1 + (0.332 - 0.913i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-3.33 - 9.16i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.49 - 0.792i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.68 - 8.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.76 - 2.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.67 + 9.52i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.723 - 4.10i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.141 + 0.0818i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.1 + 4.04i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.98 - 0.527i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.53 - 6.32i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.863 + 0.498i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.95 + 4.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.99 - 1.67i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.02 - 2.19i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + (13.2 + 2.33i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.59599834401036032558171436094, −9.905376757304395376151448886370, −9.175529114045224138962747959936, −8.613900959155349691235376599068, −7.61824113463846483490418599779, −6.66030293194985778362842869421, −5.89059599764090986309515186579, −5.20640736123708223159730725176, −3.40020005391166519608166647682, −1.48960748679092244087700108475, 0.48793425857723350146972523019, 2.23862570458669427920551016673, 2.99880175572202104513926448683, 4.23570419346859198652195170486, 5.94663408721124007163225868015, 6.88979233403659495761696831784, 7.901720677620175715350671197134, 9.167768962626204140181921629233, 9.484459406801479029891109524348, 10.43025135581688416057246267382

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