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Properties

Label 2-189-189.131-c1-0-13
Degree $2$
Conductor $189$
Sign $0.103 + 0.994i$
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  + (−0.159 − 0.190i)2-s + (−0.234 − 1.71i)3-s + (0.336 − 1.90i)4-s + (1.62 + 1.36i)5-s + (−0.289 + 0.319i)6-s + (2.64 + 0.158i)7-s + (−0.848 + 0.489i)8-s + (−2.89 + 0.803i)9-s − 0.529i·10-s + (−0.919 − 1.09i)11-s + (−3.35 − 0.130i)12-s + (−1.30 − 3.59i)13-s + (−0.392 − 0.528i)14-s + (1.96 − 3.11i)15-s + (−3.41 − 1.24i)16-s + 0.218·17-s + ⋯
L(s)  = 1  + (−0.113 − 0.134i)2-s + (−0.135 − 0.990i)3-s + (0.168 − 0.954i)4-s + (0.728 + 0.611i)5-s + (−0.118 + 0.130i)6-s + (0.998 + 0.0599i)7-s + (−0.300 + 0.173i)8-s + (−0.963 + 0.267i)9-s − 0.167i·10-s + (−0.277 − 0.330i)11-s + (−0.968 − 0.0377i)12-s + (−0.362 − 0.996i)13-s + (−0.104 − 0.141i)14-s + (0.507 − 0.804i)15-s + (−0.853 − 0.310i)16-s + 0.0529·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.103 + 0.994i$
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: \((2,\ 189,\ (\ :1/2),\ 0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921456 - 0.830533i\)
\(L(\frac12)\) \(\approx\) \(0.921456 - 0.830533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.234 + 1.71i)T \)
7 \( 1 + (-2.64 - 0.158i)T \)
good2 \( 1 + (0.159 + 0.190i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-1.62 - 1.36i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (0.919 + 1.09i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.30 + 3.59i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 0.218T + 17T^{2} \)
19 \( 1 - 2.96iT - 19T^{2} \)
23 \( 1 + (-2.64 - 7.26i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.78 - 4.90i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.81 - 0.672i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.00 + 5.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.0 + 3.66i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.29 - 7.34i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.01 + 5.78i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (12.1 - 6.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.48 - 0.540i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-4.18 + 0.738i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-11.4 - 9.58i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.29 + 2.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.13 + 1.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 - 2.50i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.141 - 0.0515i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 6.80T + 89T^{2} \)
97 \( 1 + (12.2 - 2.16i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.29255586806268190809798616611, −11.13836518104730349265904028326, −10.65066958998473862125090749010, −9.469736188802385924502971218523, −8.126503573034909738999126791940, −7.14041720533783877229535867097, −5.86802294790425458218402406988, −5.33234412173254614107936088714, −2.67996557157566168230668403773, −1.42064565857742030824624060607, 2.46102992949338142961737531257, 4.29964945004109092848712161246, 5.00343405151789091745403611410, 6.53296091419352305935633121887, 7.956219927705298476094154674627, 8.882811742403925237554584448746, 9.622635106036778777978821942160, 10.92676026623853160816964623346, 11.70584050896704756594716204708, 12.69883108567973718485247490588

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