Properties

Label 2-567-9.4-c1-0-16
Degree $2$
Conductor $567$
Sign $0.766 - 0.642i$
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  + (1.07 + 1.85i)2-s + (−1.30 + 2.25i)4-s + (1.87 − 3.24i)5-s + (−0.5 − 0.866i)7-s − 1.29·8-s + 8.03·10-s + (−0.373 − 0.646i)11-s + (3.01 − 5.22i)13-s + (1.07 − 1.85i)14-s + (1.21 + 2.10i)16-s + 0.543·17-s − 1.20·19-s + (4.87 + 8.44i)20-s + (0.800 − 1.38i)22-s + (−3.74 + 6.48i)23-s + ⋯
L(s)  = 1  + (0.758 + 1.31i)2-s + (−0.650 + 1.12i)4-s + (0.837 − 1.45i)5-s + (−0.188 − 0.327i)7-s − 0.456·8-s + 2.54·10-s + (−0.112 − 0.194i)11-s + (0.837 − 1.44i)13-s + (0.286 − 0.496i)14-s + (0.304 + 0.527i)16-s + 0.131·17-s − 0.275·19-s + (1.08 + 1.88i)20-s + (0.170 − 0.295i)22-s + (−0.781 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30116 + 0.837554i\)
\(L(\frac12)\) \(\approx\) \(2.30116 + 0.837554i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.07 - 1.85i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.87 + 3.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.373 + 0.646i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.01 + 5.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.543T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
23 \( 1 + (3.74 - 6.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.01 - 6.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (1.39 - 2.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.91 + 8.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.14 + 3.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.45T + 53T^{2} \)
59 \( 1 + (7.29 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.61 - 8.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.11 - 8.86i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (5.11 + 8.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.39 - 2.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.54T + 89T^{2} \)
97 \( 1 + (2.60 + 4.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.67487659940120941290202483599, −9.846524171321053159904380817166, −8.661119935958016349953989615282, −8.195780831051161511911699436511, −7.12146179090047228762421940973, −5.87646030596889412233419789543, −5.56786684044337190569677272556, −4.64465181446402820079628155614, −3.48070172015234276088371885998, −1.31046971586801636628892728414, 1.91384436702685900172554370375, 2.59418071420702314399250088866, 3.68174122814131003664506410270, 4.69146886866227810530422733347, 6.20422047751124216583211051268, 6.52872038723672550686979999424, 8.032929356572433146496389095368, 9.507081536643222216979444287742, 9.959671329459595025076556923630, 10.93230043844403457518701730199

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