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Properties

Label 2-189-9.4-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.368 - 0.929i$
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  + (0.849 + 1.47i)2-s + (−0.444 + 0.769i)4-s + (−1.79 + 3.10i)5-s + (0.5 + 0.866i)7-s + 1.88·8-s − 6.09·10-s + (−1.40 − 2.43i)11-s + (−0.5 + 0.866i)13-s + (−0.849 + 1.47i)14-s + (2.49 + 4.31i)16-s + 4.11·17-s − 0.888·19-s + (−1.59 − 2.76i)20-s + (2.38 − 4.13i)22-s + (2.93 − 5.08i)23-s + ⋯
L(s)  = 1  + (0.600 + 1.04i)2-s + (−0.222 + 0.384i)4-s + (−0.802 + 1.38i)5-s + (0.188 + 0.327i)7-s + 0.667·8-s − 1.92·10-s + (−0.423 − 0.733i)11-s + (−0.138 + 0.240i)13-s + (−0.227 + 0.393i)14-s + (0.623 + 1.07i)16-s + 0.997·17-s − 0.203·19-s + (−0.356 − 0.617i)20-s + (0.509 − 0.882i)22-s + (0.612 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839446 + 1.23641i\)
\(L(\frac12)\) \(\approx\) \(0.839446 + 1.23641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.849 - 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.79 - 3.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.11T + 17T^{2} \)
19 \( 1 + 0.888T + 19T^{2} \)
23 \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.849 + 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.49 + 6.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 + (4.43 - 7.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.15 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (-3.54 - 6.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.05 + 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + (3.66 + 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.18470522057810808957206757751, −11.82977032415539609169561522140, −10.97907964720295928956633918574, −10.13079144974678962761684545255, −8.324270666363428317181482010907, −7.54485292373798744737431223323, −6.61596037103994374500080168714, −5.70230577990157987499611332924, −4.29029878668166145768455608950, −2.90087674200382295163118206298, 1.37079395849934071523478881278, 3.30931639574839657528727667067, 4.51028781951976494243216193422, 5.19997247781623912807733440146, 7.40556586467059982458664509031, 8.136385171191023172850064171327, 9.485599694935336306840876028608, 10.54338980394135776316604592721, 11.60248908723861822189116365325, 12.33060537590815887426675337216

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