Properties

Label 2-189-189.88-c1-0-5
Degree $2$
Conductor $189$
Sign $0.999 - 0.0356i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (−1.23 − 1.03i)2-s + (−0.108 + 1.72i)3-s + (0.102 + 0.582i)4-s + (−0.188 + 0.157i)5-s + (1.92 − 2.01i)6-s + (2.03 − 1.69i)7-s + (−1.13 + 1.96i)8-s + (−2.97 − 0.375i)9-s + 0.395·10-s + (3.95 + 3.31i)11-s + (−1.01 + 0.114i)12-s + (4.55 + 1.65i)13-s + (−4.25 − 0.0162i)14-s + (−0.252 − 0.342i)15-s + (4.54 − 1.65i)16-s + 1.60·17-s + ⋯
L(s)  = 1  + (−0.871 − 0.731i)2-s + (−0.0626 + 0.998i)3-s + (0.0513 + 0.291i)4-s + (−0.0841 + 0.0706i)5-s + (0.784 − 0.824i)6-s + (0.768 − 0.639i)7-s + (−0.400 + 0.694i)8-s + (−0.992 − 0.125i)9-s + 0.125·10-s + (1.19 + 0.999i)11-s + (−0.293 + 0.0329i)12-s + (1.26 + 0.460i)13-s + (−1.13 − 0.00433i)14-s + (−0.0651 − 0.0884i)15-s + (1.13 − 0.413i)16-s + 0.388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.999 - 0.0356i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.999 - 0.0356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804920 + 0.0143517i\)
\(L(\frac12)\) \(\approx\) \(0.804920 + 0.0143517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.108 - 1.72i)T \)
7 \( 1 + (-2.03 + 1.69i)T \)
good2 \( 1 + (1.23 + 1.03i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.188 - 0.157i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-3.95 - 3.31i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-4.55 - 1.65i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 - 2.02T + 19T^{2} \)
23 \( 1 + (4.12 + 1.50i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-4.90 + 1.78i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.27 + 7.20i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.98 - 6.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.13 + 1.87i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.25 - 7.12i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.379 - 2.15i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.36 + 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.6 + 4.25i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.582 + 3.30i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.71 - 4.79i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.508 + 0.879i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.89 + 5.01i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.37 + 5.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.59 + 2.40i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (1.78 - 10.1i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.80054602320371369498573745821, −11.50102466037942317528359368297, −10.46547464066083177323566226871, −9.748088641039886388642800486293, −8.915102344113913896433344810394, −7.914184128656400209563322334899, −6.23952652406505502059102533339, −4.74469933690023876047967153810, −3.62508736706536902794516878348, −1.56289103057690505330169729954, 1.22420983770535996584222558852, 3.44911589215761869344343058908, 5.70605651857012503446499113312, 6.44188884281528597183017553205, 7.62627540337835139202560091096, 8.623434638838274656439087582454, 8.793247794812171789691158254103, 10.58834278264576388515470645603, 11.82060558092146786477601525941, 12.28845494489320776994453671246

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