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Properties

Label 2-189-189.131-c1-0-8
Degree $2$
Conductor $189$
Sign $0.619 + 0.785i$
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  + (−1.08 − 1.29i)2-s + (1.72 + 0.201i)3-s + (−0.151 + 0.861i)4-s + (1.13 + 0.954i)5-s + (−1.61 − 2.45i)6-s + (2.51 + 0.806i)7-s + (−1.65 + 0.953i)8-s + (2.91 + 0.693i)9-s − 2.51i·10-s + (−0.0836 − 0.0996i)11-s + (−0.435 + 1.45i)12-s + (−0.311 − 0.855i)13-s + (−1.69 − 4.15i)14-s + (1.76 + 1.87i)15-s + (4.68 + 1.70i)16-s − 5.63·17-s + ⋯
L(s)  = 1  + (−0.770 − 0.918i)2-s + (0.993 + 0.116i)3-s + (−0.0759 + 0.430i)4-s + (0.508 + 0.426i)5-s + (−0.658 − 1.00i)6-s + (0.952 + 0.304i)7-s + (−0.584 + 0.337i)8-s + (0.972 + 0.231i)9-s − 0.796i·10-s + (−0.0252 − 0.0300i)11-s + (−0.125 + 0.419i)12-s + (−0.0863 − 0.237i)13-s + (−0.453 − 1.10i)14-s + (0.455 + 0.483i)15-s + (1.17 + 0.426i)16-s − 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.619 + 0.785i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.619 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09431 - 0.530510i\)
\(L(\frac12)\) \(\approx\) \(1.09431 - 0.530510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.201i)T \)
7 \( 1 + (-2.51 - 0.806i)T \)
good2 \( 1 + (1.08 + 1.29i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-1.13 - 0.954i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (0.0836 + 0.0996i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.311 + 0.855i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 5.63T + 17T^{2} \)
19 \( 1 + 0.0959iT - 19T^{2} \)
23 \( 1 + (2.22 + 6.10i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.238 + 0.656i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (8.96 + 1.58i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-1.72 - 2.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.04 - 0.744i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.37 - 7.82i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.18 - 12.3i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.26 + 0.731i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.96 + 2.17i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.69 - 1.35i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (11.0 + 9.25i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.38 - 1.37i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.94 + 4.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.310 + 0.260i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.54 + 1.65i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (-10.2 + 1.81i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.27442853626385159834675061956, −11.05609990014616616183754328156, −10.48187076118919101940266323862, −9.424045402070332715215835840994, −8.701381926054362495900317930788, −7.79370671173691733725827452281, −6.21705635466459724088164977134, −4.51891069216117168150500758128, −2.74113380838063970774328684606, −1.88721534736917274968966389007, 1.86842985824864306189332680960, 3.87642844258091703383201254136, 5.44839869583452874166247329659, 7.00332979829701178425121906783, 7.63392040422198378088405049055, 8.795787763496253921779273014659, 9.122713574554455470153452669219, 10.34840235388617404426882658093, 11.74861207096662387168573581941, 13.03518090119010279427711783602

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