Properties

Label 2-189-21.20-c1-0-6
Degree $2$
Conductor $189$
Sign $0.377 + 0.925i$
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  − 1.41i·2-s + 1.73·5-s + (1 + 2.44i)7-s − 2.82i·8-s − 2.44i·10-s − 1.41i·11-s + 2.44i·13-s + (3.46 − 1.41i)14-s − 4.00·16-s − 5.19·17-s − 7.34i·19-s − 2.00·22-s + 2.82i·23-s − 2.00·25-s + 3.46·26-s + ⋯
L(s)  = 1  − 0.999i·2-s + 0.774·5-s + (0.377 + 0.925i)7-s − 0.999i·8-s − 0.774i·10-s − 0.426i·11-s + 0.679i·13-s + (0.925 − 0.377i)14-s − 1.00·16-s − 1.26·17-s − 1.68i·19-s − 0.426·22-s + 0.589i·23-s − 0.400·25-s + 0.679·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.377 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18979 - 0.799391i\)
\(L(\frac12)\) \(\approx\) \(1.18979 - 0.799391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2.44i)T \)
good2 \( 1 + 1.41iT - 2T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 - 2.44iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + 8.66T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 2.44iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−12.20003588939951673772072945868, −11.32445565682391576879601603432, −10.72731998292512551681967476542, −9.341091374372791732670732620300, −8.909766763618312675979689306802, −7.07752398678913708314848478297, −6.02741144301485468951493417722, −4.63727216477519667024161427335, −2.87821755845660599453994694347, −1.83112991167314617853490303370, 2.11618973299712712027780402745, 4.27294115020486697734705743889, 5.62524808166462633392988730764, 6.49192134645838242014543461484, 7.57985669344396046469323545879, 8.362479658156968219551841550469, 9.827499438068967724494874974637, 10.63960351580536385608052146590, 11.72216212277246911552028063491, 13.07651807571313266254565758175

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