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Properties

Label 2-189-7.6-c2-0-9
Degree $2$
Conductor $189$
Sign $0.866 + 0.5i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·4-s − 1.73i·5-s + (−3.5 + 6.06i)7-s + 7·8-s + 1.73i·10-s + 17·11-s − 17.3i·13-s + (3.5 − 6.06i)14-s + 5·16-s − 31.1i·17-s + 20.7i·19-s + 5.19i·20-s − 17·22-s + 32·23-s + ⋯
L(s)  = 1  − 0.5·2-s − 0.750·4-s − 0.346i·5-s + (−0.5 + 0.866i)7-s + 0.875·8-s + 0.173i·10-s + 1.54·11-s − 1.33i·13-s + (0.250 − 0.433i)14-s + 0.312·16-s − 1.83i·17-s + 1.09i·19-s + 0.259i·20-s − 0.772·22-s + 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.943206 - 0.252731i\)
\(L(\frac12)\) \(\approx\) \(0.943206 - 0.252731i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (3.5 - 6.06i)T \)
good2 \( 1 + T + 4T^{2} \)
5 \( 1 + 1.73iT - 25T^{2} \)
11 \( 1 - 17T + 121T^{2} \)
13 \( 1 + 17.3iT - 169T^{2} \)
17 \( 1 + 31.1iT - 289T^{2} \)
19 \( 1 - 20.7iT - 361T^{2} \)
23 \( 1 - 32T + 529T^{2} \)
29 \( 1 - 2T + 841T^{2} \)
31 \( 1 + 22.5iT - 961T^{2} \)
37 \( 1 - 8T + 1.36e3T^{2} \)
41 \( 1 - 13.8iT - 1.68e3T^{2} \)
43 \( 1 - 26T + 1.84e3T^{2} \)
47 \( 1 - 38.1iT - 2.20e3T^{2} \)
53 \( 1 + T + 2.80e3T^{2} \)
59 \( 1 + 90.0iT - 3.48e3T^{2} \)
61 \( 1 - 69.2iT - 3.72e3T^{2} \)
67 \( 1 + 34T + 4.48e3T^{2} \)
71 \( 1 + 10T + 5.04e3T^{2} \)
73 \( 1 - 57.1iT - 5.32e3T^{2} \)
79 \( 1 + 46T + 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 - 51.9iT - 7.92e3T^{2} \)
97 \( 1 + 91.7iT - 9.40e3T^{2} \)
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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

−12.36981925149656189851149525986, −11.28375378835216843496886879714, −9.882757929216801342078888227235, −9.244561205637630279955736772346, −8.520772940308714122307997461209, −7.26417404545643058371775978447, −5.81752683728646150907118912528, −4.72149842321488629357857362314, −3.16922775013569802381176023976, −0.894231169579700873739126580006, 1.24418870179029987255915215767, 3.68239428599772318948678333361, 4.55404610109491588348318904510, 6.48657382192593264466940966742, 7.15798989514369525373067365299, 8.759082114819729802527630746723, 9.228057589147929967056372227095, 10.39447319074812952717543262630, 11.16825109633583343081336453000, 12.51797746799365424326196893164

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