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Properties

Label 2-567-63.59-c1-0-22
Degree $2$
Conductor $567$
Sign $0.310 - 0.950i$
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  + (2.15 + 1.24i)2-s + (2.10 + 3.64i)4-s + 1.23·5-s + (2.63 + 0.248i)7-s + 5.49i·8-s + (2.66 + 1.53i)10-s − 5.49i·11-s + (−2.57 − 1.48i)13-s + (5.37 + 3.81i)14-s + (−2.63 + 4.56i)16-s + (−2.15 + 3.73i)17-s + (−4.80 + 2.77i)19-s + (2.59 + 4.5i)20-s + (6.83 − 11.8i)22-s − 1.63i·23-s + ⋯
L(s)  = 1  + (1.52 + 0.880i)2-s + (1.05 + 1.82i)4-s + 0.552·5-s + (0.995 + 0.0937i)7-s + 1.94i·8-s + (0.843 + 0.486i)10-s − 1.65i·11-s + (−0.712 − 0.411i)13-s + (1.43 + 1.01i)14-s + (−0.658 + 1.14i)16-s + (−0.523 + 0.906i)17-s + (−1.10 + 0.636i)19-s + (0.580 + 1.00i)20-s + (1.45 − 2.52i)22-s − 0.340i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.310 - 0.950i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.99737 + 2.17300i\)
\(L(\frac12)\) \(\approx\) \(2.99737 + 2.17300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.248i)T \)
good2 \( 1 + (-2.15 - 1.24i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 + 5.49iT - 11T^{2} \)
13 \( 1 + (2.57 + 1.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.15 - 3.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.80 - 2.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.63iT - 23T^{2} \)
29 \( 1 + (-0.440 + 0.254i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.04 - 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.53 - 2.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.177 - 0.306i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0318 - 0.0552i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.93 + 8.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.57 + 2.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.53 - 2.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.97 - 3.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + (6.09 + 3.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.165 + 0.287i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.69 + 6.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.71 + 2.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.85 - 2.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.15053127675477219467605241118, −10.33044793985571689480479995919, −8.653765817108996957218315230019, −8.157006364643831417221817839149, −7.04964357142442372525173809885, −5.95359966295624365994246415032, −5.58805360730848798245076340505, −4.48956522428932453711709488288, −3.51364154912462848258395544718, −2.15984944727249438565226583629, 1.91340316823924829283818106368, 2.37343074935087958811446944971, 4.16604664155371912696415340674, 4.69963307296152391294282250895, 5.50467242542907417311450747469, 6.72443573702871532540509022783, 7.58102500017492834713373710436, 9.232220817522408577692815672627, 9.936098144282836584716498774565, 10.95934295486037659547203519960

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