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Properties

Label 2-567-189.101-c1-0-5
Degree $2$
Conductor $567$
Sign $-0.733 - 0.679i$
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  + (−0.594 + 0.708i)2-s + (0.198 + 1.12i)4-s + (0.386 − 0.324i)5-s + (0.529 − 2.59i)7-s + (−2.51 − 1.45i)8-s + 0.466i·10-s + (−3.12 + 3.72i)11-s + (−1.60 + 4.42i)13-s + (1.52 + 1.91i)14-s + (0.374 − 0.136i)16-s + 4.77·17-s + 6.37i·19-s + (0.442 + 0.371i)20-s + (−0.780 − 4.42i)22-s + (−1.22 + 3.37i)23-s + ⋯
L(s)  = 1  + (−0.420 + 0.500i)2-s + (0.0994 + 0.563i)4-s + (0.172 − 0.144i)5-s + (0.200 − 0.979i)7-s + (−0.890 − 0.514i)8-s + 0.147i·10-s + (−0.941 + 1.12i)11-s + (−0.446 + 1.22i)13-s + (0.406 + 0.512i)14-s + (0.0936 − 0.0341i)16-s + 1.15·17-s + 1.46i·19-s + (0.0989 + 0.0830i)20-s + (−0.166 − 0.943i)22-s + (−0.255 + 0.702i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.733 - 0.679i$
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: \((2,\ 567,\ (\ :1/2),\ -0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325476 + 0.830689i\)
\(L(\frac12)\) \(\approx\) \(0.325476 + 0.830689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.529 + 2.59i)T \)
good2 \( 1 + (0.594 - 0.708i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.386 + 0.324i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (3.12 - 3.72i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.60 - 4.42i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 - 6.37iT - 19T^{2} \)
23 \( 1 + (1.22 - 3.37i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.997 + 2.73i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.773 - 0.136i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.73 + 4.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.44 + 1.98i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.36 - 7.75i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.33 - 7.56i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.26 - 2.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.01 - 1.82i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.07 + 1.24i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.70 + 5.62i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-11.3 + 6.53i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.42 + 4.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.63 + 3.05i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.75 - 1.00i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (2.41 + 0.426i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.02147593589203843386170804478, −9.783555597776210974555982329377, −9.582706762018996445198405545238, −8.043585168478078583260581452917, −7.62769468888382963483279719446, −6.92043948930950344279903013777, −5.69081152939933296609639774332, −4.44825574297000420093063267483, −3.46413870806712663169396114249, −1.81745804520843947358170618777, 0.56576954116813990769640818519, 2.39939942903388205163841222197, 3.05834460425796428342051994408, 5.25978620885747246993679789833, 5.49898126600000060256212822527, 6.67140729075990780264056610705, 8.169235505897239041910074928447, 8.598523993220113807854977842403, 9.784303533219277122161262327621, 10.35863577379012414591208280170

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