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Properties

Label 2-567-189.101-c1-0-10
Degree $2$
Conductor $567$
Sign $0.945 - 0.324i$
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  + (−0.905 + 1.07i)2-s + (0.00302 + 0.0171i)4-s + (−1.61 + 1.35i)5-s + (1.08 − 2.41i)7-s + (−2.46 − 1.42i)8-s − 2.97i·10-s + (2.38 − 2.83i)11-s + (2.27 − 6.25i)13-s + (1.61 + 3.35i)14-s + (3.72 − 1.35i)16-s + 3.23·17-s + 5.32i·19-s + (−0.0281 − 0.0236i)20-s + (0.905 + 5.13i)22-s + (0.239 − 0.658i)23-s + ⋯
L(s)  = 1  + (−0.639 + 0.762i)2-s + (0.00151 + 0.00857i)4-s + (−0.722 + 0.606i)5-s + (0.411 − 0.911i)7-s + (−0.869 − 0.502i)8-s − 0.939i·10-s + (0.717 − 0.855i)11-s + (0.631 − 1.73i)13-s + (0.432 + 0.897i)14-s + (0.931 − 0.339i)16-s + 0.785·17-s + 1.22i·19-s + (−0.00629 − 0.00528i)20-s + (0.193 + 1.09i)22-s + (0.0499 − 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922350 + 0.153949i\)
\(L(\frac12)\) \(\approx\) \(0.922350 + 0.153949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 + 2.41i)T \)
good2 \( 1 + (0.905 - 1.07i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.61 - 1.35i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.38 + 2.83i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.27 + 6.25i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 - 5.32iT - 19T^{2} \)
23 \( 1 + (-0.239 + 0.658i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.50 + 4.12i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.47 - 0.260i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.24 - 5.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.4 - 3.81i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.29 + 7.32i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.04 + 5.90i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.766 - 0.442i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.85 + 2.13i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.32 - 1.29i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.05 + 6.76i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-3.83 + 2.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.65 - 1.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.29 + 6.11i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-11.4 + 4.16i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 + (-6.85 - 1.20i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.68384826426346698690322910416, −9.941503858620081794072678173233, −8.665688489747038566874826804495, −7.86540193028989923754895349961, −7.59848506200019926338106205788, −6.44219588488736744767173800643, −5.60678238227654130258602882892, −3.73989930335897335244260229974, −3.36393823428496119847240730100, −0.789387894684552187822740186593, 1.28099267870412349006290835261, 2.35728591116382479493759175902, 3.96714824361674971033022909672, 4.95433098244242461716990835261, 6.12936963919878333098731764510, 7.25849766172309170986545007128, 8.472543170082636287624819175570, 9.170425614488302163819261459408, 9.486960145377153987077605359113, 10.96638735491963879949181206086

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