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Properties

Label 2-189-63.47-c1-0-3
Degree $2$
Conductor $189$
Sign $0.991 - 0.133i$
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  + (0.621 − 0.359i)2-s + (−0.742 + 1.28i)4-s + 1.44·5-s + (2.19 − 1.47i)7-s + 2.50i·8-s + (0.900 − 0.519i)10-s + 1.80i·11-s + (1.88 − 1.09i)13-s + (0.834 − 1.70i)14-s + (−0.585 − 1.01i)16-s + (1.95 + 3.38i)17-s + (−3.47 − 2.00i)19-s + (−1.07 + 1.86i)20-s + (0.646 + 1.11i)22-s − 5.67i·23-s + ⋯
L(s)  = 1  + (0.439 − 0.253i)2-s + (−0.371 + 0.642i)4-s + 0.647·5-s + (0.829 − 0.558i)7-s + 0.884i·8-s + (0.284 − 0.164i)10-s + 0.542i·11-s + (0.523 − 0.302i)13-s + (0.222 − 0.456i)14-s + (−0.146 − 0.253i)16-s + (0.473 + 0.820i)17-s + (−0.797 − 0.460i)19-s + (−0.240 + 0.416i)20-s + (0.137 + 0.238i)22-s − 1.18i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53361 + 0.102963i\)
\(L(\frac12)\) \(\approx\) \(1.53361 + 0.102963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 1.47i)T \)
good2 \( 1 + (-0.621 + 0.359i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (-1.88 + 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.95 - 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.47 + 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.67iT - 23T^{2} \)
29 \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.90 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.16 + 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.996 + 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.89 + 8.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.03 + 1.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.156 + 0.270i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (-2.42 + 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.21 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.60 + 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.28 - 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 - 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.94093481898094710389448321481, −11.63348029997864830491155857673, −10.80123968638069696807973380418, −9.675373569384598994750911613358, −8.425086591489759498892702672192, −7.67147150456442409417283835515, −6.13346283811520041140315191935, −4.81912146376243006071843865895, −3.84047074365817117871382740445, −2.09670861751888765707811504253, 1.73982669211735289751978603091, 3.86184859608808813795456968732, 5.41019459028805771785899022615, 5.76749749144168883934139875804, 7.25347701382970651203916844699, 8.724766359233091341119128576323, 9.459078189237520402551391139683, 10.61357511172904751203464322702, 11.53799258248133023122553796446, 12.76999512342218699134397704103

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