Properties

Label 2-189-189.101-c1-0-4
Degree $2$
Conductor $189$
Sign $0.311 - 0.950i$
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  + (0.572 − 0.682i)2-s + (−1.73 + 0.0474i)3-s + (0.209 + 1.18i)4-s + (−1.06 + 0.891i)5-s + (−0.959 + 1.20i)6-s + (−1.58 + 2.11i)7-s + (2.47 + 1.42i)8-s + (2.99 − 0.164i)9-s + 1.23i·10-s + (−0.371 + 0.443i)11-s + (−0.418 − 2.04i)12-s + (−1.85 + 5.11i)13-s + (0.539 + 2.29i)14-s + (1.79 − 1.59i)15-s + (0.126 − 0.0459i)16-s + 2.31·17-s + ⋯
L(s)  = 1  + (0.405 − 0.482i)2-s + (−0.999 + 0.0273i)3-s + (0.104 + 0.593i)4-s + (−0.474 + 0.398i)5-s + (−0.391 + 0.493i)6-s + (−0.598 + 0.800i)7-s + (0.874 + 0.505i)8-s + (0.998 − 0.0547i)9-s + 0.390i·10-s + (−0.112 + 0.133i)11-s + (−0.120 − 0.590i)12-s + (−0.515 + 1.41i)13-s + (0.144 + 0.613i)14-s + (0.463 − 0.411i)15-s + (0.0315 − 0.0114i)16-s + 0.560·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729235 + 0.528424i\)
\(L(\frac12)\) \(\approx\) \(0.729235 + 0.528424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0474i)T \)
7 \( 1 + (1.58 - 2.11i)T \)
good2 \( 1 + (-0.572 + 0.682i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.06 - 0.891i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.371 - 0.443i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.85 - 5.11i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 2.31T + 17T^{2} \)
19 \( 1 + 3.42iT - 19T^{2} \)
23 \( 1 + (-2.28 + 6.28i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.224 + 0.616i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.71 + 0.302i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.542 + 0.939i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.37 - 0.500i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.681 - 3.86i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.07 + 6.06i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-8.16 - 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.76 - 1.73i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-13.3 - 2.34i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.31 - 5.29i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-13.6 + 7.86i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.07 + 1.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.05 + 1.72i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (11.2 - 4.10i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-18.3 - 3.23i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.49320752424240200160687856524, −11.77708942752717316538310555364, −11.25167198177834607354394072593, −10.07564259571128733809946211623, −8.853296498092160072328179536867, −7.33545998598829267015979219390, −6.58978408643786738343192865936, −5.08268490781995568486002266254, −3.98280348170813878393351504437, −2.48349064659820542076239665373, 0.830965626818588458410407700782, 3.82234930510643364110611132616, 5.10442220927966278653714538522, 5.86471971153876028625061269390, 7.04954213722341718450141333201, 7.85975654216662768893877923017, 9.886752748747586068319280305396, 10.29941373364756987904378546957, 11.38849974353417228603966001802, 12.55474226673230885320049640504

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