Properties

Label 2-192-3.2-c10-0-50
Degree $2$
Conductor $192$
Sign $0.824 + 0.566i$
Analytic cond. $121.988$
Root an. cond. $11.0448$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−200. − 137. i)3-s + 3.63e3i·5-s + 2.32e4·7-s + (2.11e4 + 5.51e4i)9-s + 6.24e4i·11-s + 1.70e5·13-s + (4.99e5 − 7.27e5i)15-s − 2.66e6i·17-s + 7.66e5·19-s + (−4.65e6 − 3.19e6i)21-s − 1.40e6i·23-s − 3.41e6·25-s + (3.34e6 − 1.39e7i)27-s − 4.83e6i·29-s + 4.18e7·31-s + ⋯
L(s)  = 1  + (−0.824 − 0.566i)3-s + 1.16i·5-s + 1.38·7-s + (0.358 + 0.933i)9-s + 0.387i·11-s + 0.458·13-s + (0.657 − 0.957i)15-s − 1.87i·17-s + 0.309·19-s + (−1.13 − 0.782i)21-s − 0.218i·23-s − 0.349·25-s + (0.233 − 0.972i)27-s − 0.235i·29-s + 1.46·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.824 + 0.566i$
Analytic conductor: \(121.988\)
Root analytic conductor: \(11.0448\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :5),\ 0.824 + 0.566i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.997243454\)
\(L(\frac12)\) \(\approx\) \(1.997243454\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (200. + 137. i)T \)
good5 \( 1 - 3.63e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.32e4T + 2.82e8T^{2} \)
11 \( 1 - 6.24e4iT - 2.59e10T^{2} \)
13 \( 1 - 1.70e5T + 1.37e11T^{2} \)
17 \( 1 + 2.66e6iT - 2.01e12T^{2} \)
19 \( 1 - 7.66e5T + 6.13e12T^{2} \)
23 \( 1 + 1.40e6iT - 4.14e13T^{2} \)
29 \( 1 + 4.83e6iT - 4.20e14T^{2} \)
31 \( 1 - 4.18e7T + 8.19e14T^{2} \)
37 \( 1 + 5.01e7T + 4.80e15T^{2} \)
41 \( 1 + 1.49e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.98e8T + 2.16e16T^{2} \)
47 \( 1 - 1.55e8iT - 5.25e16T^{2} \)
53 \( 1 + 4.21e7iT - 1.74e17T^{2} \)
59 \( 1 - 2.92e8iT - 5.11e17T^{2} \)
61 \( 1 - 5.30e8T + 7.13e17T^{2} \)
67 \( 1 - 5.22e8T + 1.82e18T^{2} \)
71 \( 1 + 5.71e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.18e9T + 4.29e18T^{2} \)
79 \( 1 + 1.96e9T + 9.46e18T^{2} \)
83 \( 1 + 2.18e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.38e8iT - 3.11e19T^{2} \)
97 \( 1 + 8.84e9T + 7.37e19T^{2} \)
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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

−10.88964668190989640747117188308, −9.922541713103025994284676183028, −8.327405013977309521493744429408, −7.32665656466046826236131285085, −6.72305272954667597238314049383, −5.41734753088396967498813547916, −4.55787359187039846350009073756, −2.83007677399952487012573092861, −1.76115279320876141605324021915, −0.58123278671160711818418598925, 0.927543187069921424963790514847, 1.57833705181264508389992955616, 3.72754170224257722961055606495, 4.70139759589804764574248632708, 5.36939854324026970416066476663, 6.43245027789842345029052175845, 8.177632367459978978261183245544, 8.611231984976783490433659982828, 9.942767134437899599542994623760, 10.90531547772054789706196495803

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