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Properties

Label 2-189-189.131-c1-0-9
Degree $2$
Conductor $189$
Sign $-0.449 - 0.893i$
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  + (1.34 + 1.60i)2-s + (0.0648 + 1.73i)3-s + (−0.412 + 2.34i)4-s + (0.275 + 0.231i)5-s + (−2.68 + 2.43i)6-s + (2.18 − 1.48i)7-s + (−0.682 + 0.393i)8-s + (−2.99 + 0.224i)9-s + 0.753i·10-s + (−4.02 − 4.79i)11-s + (−4.07 − 0.562i)12-s + (−0.429 − 1.18i)13-s + (5.32 + 1.50i)14-s + (−0.382 + 0.492i)15-s + (2.91 + 1.06i)16-s − 0.937·17-s + ⋯
L(s)  = 1  + (0.950 + 1.13i)2-s + (0.0374 + 0.999i)3-s + (−0.206 + 1.17i)4-s + (0.123 + 0.103i)5-s + (−1.09 + 0.992i)6-s + (0.826 − 0.562i)7-s + (−0.241 + 0.139i)8-s + (−0.997 + 0.0748i)9-s + 0.238i·10-s + (−1.21 − 1.44i)11-s + (−1.17 − 0.162i)12-s + (−0.119 − 0.327i)13-s + (1.42 + 0.402i)14-s + (−0.0988 + 0.127i)15-s + (0.729 + 0.265i)16-s − 0.227·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.449 - 0.893i$
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: \((2,\ 189,\ (\ :1/2),\ -0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999894 + 1.62285i\)
\(L(\frac12)\) \(\approx\) \(0.999894 + 1.62285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0648 - 1.73i)T \)
7 \( 1 + (-2.18 + 1.48i)T \)
good2 \( 1 + (-1.34 - 1.60i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.275 - 0.231i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (4.02 + 4.79i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.429 + 1.18i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 0.937T + 17T^{2} \)
19 \( 1 - 7.64iT - 19T^{2} \)
23 \( 1 + (0.584 + 1.60i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.731 - 2.00i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.71 - 1.00i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-1.66 - 2.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.66 - 3.15i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.688 + 3.90i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.876 - 4.96i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.01 - 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.00 + 2.18i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (10.3 - 1.81i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.222 + 0.186i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.18 + 2.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.19 + 1.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.90 + 6.63i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.06 - 2.20i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + (7.91 - 1.39i)T + (91.1 - 33.1i)T^{2} \)
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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

−13.45560645780329636845273176711, −12.02191921884807073847841468139, −10.68672317346131645756414247450, −10.24025887286840814169016381775, −8.308032526988853205876890354183, −7.957146340466866956713972284117, −6.23468009336392138126580024874, −5.40536287791748421031166734680, −4.48166504837461339191298881305, −3.28032405396210625880611920173, 1.87886077041601464381543998017, 2.68069300887197569354552504154, 4.66995675603871063238523977797, 5.40238711042154922675220161073, 7.08324249926027893509945272360, 8.067818510300314053208373596590, 9.440945731418004088750740145775, 10.77427388189753592642336534957, 11.62107751493246212924509815194, 12.24150341069359170268976942391

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