Properties

Label 2-567-189.142-c1-0-12
Degree $2$
Conductor $567$
Sign $0.0581 + 0.998i$
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  + (0.279 − 1.58i)2-s + (−0.555 − 0.202i)4-s + (−0.230 − 0.0840i)5-s + (−1.31 + 2.29i)7-s + (1.13 − 1.96i)8-s + (−0.197 + 0.342i)10-s + (4.84 − 1.76i)11-s + (4.55 + 1.65i)13-s + (3.27 + 2.72i)14-s + (−3.70 − 3.10i)16-s + (0.800 − 1.38i)17-s + (−1.01 − 1.75i)19-s + (0.111 + 0.0933i)20-s + (−1.44 − 8.17i)22-s + (−0.761 − 4.32i)23-s + ⋯
L(s)  = 1  + (0.197 − 1.12i)2-s + (−0.277 − 0.101i)4-s + (−0.103 − 0.0375i)5-s + (−0.496 + 0.867i)7-s + (0.400 − 0.694i)8-s + (−0.0625 + 0.108i)10-s + (1.46 − 0.531i)11-s + (1.26 + 0.460i)13-s + (0.874 + 0.728i)14-s + (−0.925 − 0.776i)16-s + (0.194 − 0.336i)17-s + (−0.232 − 0.403i)19-s + (0.0248 + 0.0208i)20-s + (−0.307 − 1.74i)22-s + (−0.158 − 0.900i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.0581 + 0.998i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.0581 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33174 - 1.25644i\)
\(L(\frac12)\) \(\approx\) \(1.33174 - 1.25644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 - 2.29i)T \)
good2 \( 1 + (-0.279 + 1.58i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.230 + 0.0840i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-4.84 + 1.76i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-4.55 - 1.65i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.800 + 1.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.01 + 1.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.761 + 4.32i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (4.90 - 1.78i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.87 - 2.50i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 - 7.97T + 37T^{2} \)
41 \( 1 + (-5.13 - 1.87i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.25 - 7.12i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.05 - 0.748i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (3.36 + 5.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.52 - 7.99i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (3.15 - 1.14i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.29 + 7.34i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.508 - 0.879i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.79T + 73T^{2} \)
79 \( 1 + (1.44 - 8.19i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.59 - 2.40i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-7.78 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.78 - 10.1i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.84272481486577139871686619681, −9.632775192854907017627793598519, −9.115336868103360180443495748758, −8.119953470051880848623705707644, −6.55260257437120772379611790285, −6.17575645203606416241747755843, −4.45685674354426748975741830350, −3.60170584044993088556485827133, −2.58375022792464406016137811857, −1.21422879365004700666537617900, 1.52455802013513650101386969081, 3.61355915799305174162827483785, 4.34524523588699002145631985711, 5.91257613333457881325516391560, 6.29101604843298287808598894656, 7.34752303439965600988695142413, 7.935562281792512132848463326215, 9.082722500132745703949701111898, 9.967759473770073241117369363167, 11.01867307227845534661470802062

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