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Properties

Label 2-567-189.104-c1-0-6
Degree $2$
Conductor $567$
Sign $-0.712 - 0.701i$
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  + (−0.735 + 2.01i)2-s + (−2.00 − 1.68i)4-s + (−0.0677 + 0.384i)5-s + (0.618 − 2.57i)7-s + (1.15 − 0.666i)8-s + (−0.726 − 0.419i)10-s + (4.77 − 0.842i)11-s + (1.96 + 5.38i)13-s + (4.74 + 3.14i)14-s + (−0.412 − 2.34i)16-s + (−3.57 + 6.18i)17-s + (0.398 − 0.230i)19-s + (0.783 − 0.657i)20-s + (−1.81 + 10.2i)22-s + (−0.358 + 0.427i)23-s + ⋯
L(s)  = 1  + (−0.519 + 1.42i)2-s + (−1.00 − 0.841i)4-s + (−0.0303 + 0.171i)5-s + (0.233 − 0.972i)7-s + (0.407 − 0.235i)8-s + (−0.229 − 0.132i)10-s + (1.44 − 0.254i)11-s + (0.543 + 1.49i)13-s + (1.26 + 0.839i)14-s + (−0.103 − 0.585i)16-s + (−0.866 + 1.50i)17-s + (0.0914 − 0.0528i)19-s + (0.175 − 0.146i)20-s + (−0.386 + 2.18i)22-s + (−0.0748 + 0.0891i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416876 + 1.01800i\)
\(L(\frac12)\) \(\approx\) \(0.416876 + 1.01800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 2.57i)T \)
good2 \( 1 + (0.735 - 2.01i)T + (-1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.0677 - 0.384i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-4.77 + 0.842i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-1.96 - 5.38i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.398 + 0.230i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.358 - 0.427i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.73 - 4.76i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.14 + 2.55i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (1.07 - 1.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.917 - 0.334i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.16 - 6.58i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.83 + 2.38i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 3.13iT - 53T^{2} \)
59 \( 1 + (-0.715 + 4.05i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.89 - 9.40i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-6.30 + 2.29i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.08 + 3.51i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.55 - 3.78i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.818 + 0.297i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.78 + 2.46i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (2.65 + 4.60i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 1.90i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.05816136859883761040602296094, −9.909925527169745741543399415815, −8.889660270545762900813884119167, −8.531161575552021775393171100640, −7.26282465278382003381717603134, −6.67750598444253384188269737986, −6.11057855156938143458134136352, −4.58854495564319529378510353079, −3.76355230256479851157594157765, −1.44157148906187507135238956422, 0.857688289764358318450164057563, 2.26661666186305116881927801581, 3.22649503390756061791570036866, 4.44491848169379887957991118653, 5.68417872914112567540052555841, 6.83443179480046364569685672590, 8.269997810592088135470649620936, 8.956951250348824561933482973252, 9.526965819031687469764488706104, 10.51417648412355542177949661966

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