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Properties

Label 2-567-63.4-c1-0-17
Degree $2$
Conductor $567$
Sign $0.884 + 0.466i$
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  + (0.635 − 1.10i)2-s + (0.193 + 0.334i)4-s + 1.55·5-s + (2.63 + 0.193i)7-s + 3.03·8-s + (0.986 − 1.70i)10-s − 3.21·11-s + (−2.39 + 4.14i)13-s + (1.88 − 2.77i)14-s + (1.53 − 2.66i)16-s + (1.05 − 1.83i)17-s + (2.43 + 4.21i)19-s + (0.300 + 0.520i)20-s + (−2.03 + 3.53i)22-s + 3.70·23-s + ⋯
L(s)  = 1  + (0.449 − 0.777i)2-s + (0.0966 + 0.167i)4-s + 0.694·5-s + (0.997 + 0.0730i)7-s + 1.07·8-s + (0.311 − 0.540i)10-s − 0.967·11-s + (−0.663 + 1.14i)13-s + (0.504 − 0.742i)14-s + (0.384 − 0.666i)16-s + (0.256 − 0.444i)17-s + (0.557 + 0.966i)19-s + (0.0671 + 0.116i)20-s + (−0.434 + 0.752i)22-s + 0.773·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31076 - 0.572236i\)
\(L(\frac12)\) \(\approx\) \(2.31076 - 0.572236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.193i)T \)
good2 \( 1 + (-0.635 + 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.55T + 5T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 + (2.39 - 4.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.05 + 1.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 4.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.70T + 23T^{2} \)
29 \( 1 + (3.68 + 6.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.75 + 4.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0932 - 0.161i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.39 + 9.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.43 + 4.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.885 - 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.834 - 1.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.91 + 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 - 5.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.11 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (5.93 - 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.654 + 1.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.70 - 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.28 - 9.15i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83513295008178250106146868065, −10.00796663480536279814836859975, −9.131287339161121828707242688993, −7.78429146442670630366390971370, −7.37532814255408657901354398673, −5.79133148967086926755918325132, −4.93212284541547181612092442286, −3.95111947793041938800868340489, −2.50743919656148616054693632932, −1.77583265404332706531912237190, 1.50343697198153414484238088087, 2.90721989834246751336745260361, 4.80344449059157582358302515305, 5.22997395420830242878933449854, 6.04258384368626212474747903793, 7.36345740527479585321822268212, 7.73515598146990685322067161683, 8.952618127550566438965256524126, 10.16748542641715685058962976579, 10.65089139613912301716942343043

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