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Properties

Label 2-567-189.131-c1-0-21
Degree $2$
Conductor $567$
Sign $-0.911 - 0.410i$
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  + (0.0103 + 0.0123i)2-s + (0.347 − 1.96i)4-s + (−3.05 − 2.56i)5-s + (−2.57 − 0.614i)7-s + (0.0559 − 0.0323i)8-s − 0.0645i·10-s + (3.31 + 3.95i)11-s + (0.220 + 0.606i)13-s + (−0.0191 − 0.0382i)14-s + (−3.75 − 1.36i)16-s − 3.38·17-s + 0.379i·19-s + (−6.11 + 5.13i)20-s + (−0.0144 + 0.0821i)22-s + (−0.344 − 0.947i)23-s + ⋯
L(s)  = 1  + (0.00734 + 0.00875i)2-s + (0.173 − 0.984i)4-s + (−1.36 − 1.14i)5-s + (−0.972 − 0.232i)7-s + (0.0197 − 0.0114i)8-s − 0.0204i·10-s + (0.999 + 1.19i)11-s + (0.0612 + 0.168i)13-s + (−0.00511 − 0.0102i)14-s + (−0.939 − 0.341i)16-s − 0.821·17-s + 0.0870i·19-s + (−1.36 + 1.14i)20-s + (−0.00308 + 0.0175i)22-s + (−0.0719 − 0.197i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.911 - 0.410i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0701306 + 0.326625i\)
\(L(\frac12)\) \(\approx\) \(0.0701306 + 0.326625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 + 0.614i)T \)
good2 \( 1 + (-0.0103 - 0.0123i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (3.05 + 2.56i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-3.31 - 3.95i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.220 - 0.606i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 - 0.379iT - 19T^{2} \)
23 \( 1 + (0.344 + 0.947i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.97 - 5.41i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (6.35 + 1.12i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.08 + 5.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.266 + 0.0969i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.45 + 8.22i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.27 + 7.24i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.71 + 0.992i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.21 + 1.17i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.93 - 1.39i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.23 - 6.06i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (8.57 + 4.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.13 + 3.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.980 - 0.823i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.07 + 2.93i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 5.35T + 89T^{2} \)
97 \( 1 + (-7.39 + 1.30i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.21082387922841492675276304508, −9.179531944147350886570914747532, −8.846198466844211219172428730728, −7.28112689434491721719914680536, −6.84072898581367555963188941848, −5.49882129078266142641290532998, −4.46299792503860618028998357078, −3.76110098074293437054289395053, −1.71462161552364589182595920795, −0.18581588897146692395456655241, 2.80845936926166808884083525326, 3.47999158989174448932328116437, 4.16417146330274999539201618265, 6.19949680351202581990208108831, 6.78099932866780653894489788872, 7.64719714507437860270730362845, 8.475928569641058985601280495782, 9.320898091711893007745417210604, 10.68277714709257576100929212257, 11.43107945498833775039333163568

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