Properties

Label 2-567-7.2-c1-0-13
Degree $2$
Conductor $567$
Sign $0.999 - 0.0109i$
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  + (−1.19 + 2.06i)2-s + (−1.84 − 3.20i)4-s + (−1.46 + 2.52i)5-s + (−2.21 − 1.44i)7-s + 4.05·8-s + (−3.48 − 6.03i)10-s + (−0.676 − 1.17i)11-s + 1.46·13-s + (5.62 − 2.86i)14-s + (−1.13 + 1.96i)16-s + (−1.65 − 2.86i)17-s + (−1.10 + 1.91i)19-s + 10.7·20-s + 3.23·22-s + (1.31 − 2.27i)23-s + ⋯
L(s)  = 1  + (−0.843 + 1.46i)2-s + (−0.924 − 1.60i)4-s + (−0.653 + 1.13i)5-s + (−0.838 − 0.544i)7-s + 1.43·8-s + (−1.10 − 1.90i)10-s + (−0.204 − 0.353i)11-s + 0.406·13-s + (1.50 − 0.766i)14-s + (−0.284 + 0.492i)16-s + (−0.401 − 0.695i)17-s + (−0.253 + 0.438i)19-s + 2.41·20-s + 0.688·22-s + (0.274 − 0.474i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.999 - 0.0109i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.999 - 0.0109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.353219 + 0.00193927i\)
\(L(\frac12)\) \(\approx\) \(0.353219 + 0.00193927i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 1.44i)T \)
good2 \( 1 + (1.19 - 2.06i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.676 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + (1.65 + 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.10 - 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
43 \( 1 - 4.34T + 43T^{2} \)
47 \( 1 + (-1.98 + 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.22 - 5.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.279 - 0.484i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.96T + 83T^{2} \)
89 \( 1 + (3.20 - 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.28T + 97T^{2} \)
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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

−10.58696846710908659285567943298, −9.658415300221324354315516677452, −8.842671944138517331162752141979, −7.79064759859566333201547128894, −7.21047383666032385508765446149, −6.51574341579015133406176003808, −5.74459139206124146180399425298, −4.18332359962104944186041878387, −2.97085593764786714275405364153, −0.30289335633676641798923121586, 1.21230099400409468048425297378, 2.64068758623870610240380326289, 3.75363418926082418244506043854, 4.73818255344292903025941406861, 6.17351675996762658021460429435, 7.63145172029659964370924394262, 8.695819717256752937303839563485, 8.920226138185292133383963489401, 9.901336642158352946355156719185, 10.68986781844025102250332977376

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