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Properties

Label 2-189-189.184-c1-0-19
Degree $2$
Conductor $189$
Sign $-0.991 + 0.129i$
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  + (−0.402 − 2.28i)2-s + (1.40 − 1.00i)3-s + (−3.16 + 1.15i)4-s + (0.125 − 0.711i)5-s + (−2.86 − 2.80i)6-s + (−2.42 + 1.05i)7-s + (1.57 + 2.73i)8-s + (0.961 − 2.84i)9-s − 1.67·10-s + (−0.324 − 1.83i)11-s + (−3.28 + 4.81i)12-s + (−1.90 − 1.59i)13-s + (3.38 + 5.10i)14-s + (−0.541 − 1.12i)15-s + (0.451 − 0.378i)16-s + 6.86·17-s + ⋯
L(s)  = 1  + (−0.284 − 1.61i)2-s + (0.812 − 0.582i)3-s + (−1.58 + 0.575i)4-s + (0.0560 − 0.318i)5-s + (−1.17 − 1.14i)6-s + (−0.916 + 0.399i)7-s + (0.558 + 0.967i)8-s + (0.320 − 0.947i)9-s − 0.528·10-s + (−0.0977 − 0.554i)11-s + (−0.948 + 1.38i)12-s + (−0.528 − 0.443i)13-s + (0.904 + 1.36i)14-s + (−0.139 − 0.291i)15-s + (0.112 − 0.0946i)16-s + 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0732108 - 1.12199i\)
\(L(\frac12)\) \(\approx\) \(0.0732108 - 1.12199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.00i)T \)
7 \( 1 + (2.42 - 1.05i)T \)
good2 \( 1 + (0.402 + 2.28i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.125 + 0.711i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.324 + 1.83i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.90 + 1.59i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 - 6.86T + 17T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
23 \( 1 + (-2.68 - 2.25i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.282 - 0.237i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (7.40 - 2.69i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.35 - 2.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.493 + 0.413i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.01 - 1.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.67 - 0.975i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.24 + 7.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.47 + 5.43i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (4.38 + 1.59i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.00 - 11.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.04 + 6.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.29 - 12.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.88 - 16.3i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (12.8 - 10.7i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 + (1.25 + 0.458i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.41985233993162170263101889712, −11.22946626746541799708932213322, −9.881165982741690093963573904874, −9.435641084073127392984616921806, −8.446420032447295782671108191833, −7.22675401226397741964888243123, −5.49664872614095387977101892071, −3.45075512832650405415896712128, −2.89801325939198025888450449501, −1.14663513282557014150541146308, 3.12582043458981857894235047175, 4.63076219942679903604633566235, 5.83883123684158945427233473393, 7.28404924637059464589670238164, 7.55838187629683980102400097697, 9.065629605430881554803536622662, 9.611159338579340984236939050746, 10.53323122592925289879205545748, 12.37029774371946893933252930079, 13.55545669825734450554843339576

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