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Properties

Label 2-567-189.101-c1-0-15
Degree $2$
Conductor $567$
Sign $0.892 + 0.451i$
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  + (0.720 − 0.858i)2-s + (0.129 + 0.731i)4-s + (3.14 − 2.63i)5-s + (0.864 + 2.50i)7-s + (2.66 + 1.53i)8-s − 4.59i·10-s + (0.647 − 0.771i)11-s + (−1.20 + 3.32i)13-s + (2.77 + 1.05i)14-s + (1.84 − 0.670i)16-s − 6.15·17-s + 2.40i·19-s + (2.33 + 1.95i)20-s + (−0.196 − 1.11i)22-s + (0.696 − 1.91i)23-s + ⋯
L(s)  = 1  + (0.509 − 0.607i)2-s + (0.0645 + 0.365i)4-s + (1.40 − 1.17i)5-s + (0.326 + 0.945i)7-s + (0.941 + 0.543i)8-s − 1.45i·10-s + (0.195 − 0.232i)11-s + (−0.335 + 0.921i)13-s + (0.740 + 0.283i)14-s + (0.460 − 0.167i)16-s − 1.49·17-s + 0.552i·19-s + (0.521 + 0.437i)20-s + (−0.0418 − 0.237i)22-s + (0.145 − 0.398i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.892 + 0.451i$
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: \((2,\ 567,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43963 - 0.582410i\)
\(L(\frac12)\) \(\approx\) \(2.43963 - 0.582410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.864 - 2.50i)T \)
good2 \( 1 + (-0.720 + 0.858i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-3.14 + 2.63i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.647 + 0.771i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.20 - 3.32i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 6.15T + 17T^{2} \)
19 \( 1 - 2.40iT - 19T^{2} \)
23 \( 1 + (-0.696 + 1.91i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.707 + 1.94i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.819 + 0.144i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.98 + 3.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.37 + 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.78 + 10.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.26 + 7.16i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.60 - 2.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.60 + 0.949i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.70 - 0.652i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.49 - 1.25i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.21 - 3.00i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.29 - 3.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 - 4.72i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-7.39 + 2.69i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (5.98 + 1.05i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.82333336485423336217262330378, −9.760741459606329677470271465592, −8.782927728647905368099012920715, −8.543275419803911881238154433448, −6.91661359566727617122904671662, −5.79502444969149133843520989840, −4.97875686134909816213161543643, −4.14007990341592826910364710612, −2.36435639997985660095997061640, −1.84013316246325464278146622516, 1.59782000530170614501307051586, 2.92224204644045117905063536047, 4.47662031073078749481161152622, 5.37147738061416241547230647176, 6.44494479595967902355366792722, 6.82186631966153375789893640242, 7.74646685519338093753671609992, 9.334821361195240533149227254076, 10.08884574178941985688817525451, 10.68898544092095117498231746700

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