Properties

Label 2-192-4.3-c8-0-7
Degree $2$
Conductor $192$
Sign $-1$
Analytic cond. $78.2166$
Root an. cond. $8.84402$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 46.7i·3-s + 149.·5-s + 3.20e3i·7-s − 2.18e3·9-s + 3.23e3i·11-s − 176.·13-s + 6.98e3i·15-s + 8.04e4·17-s + 2.39e5i·19-s − 1.50e5·21-s + 8.53e4i·23-s − 3.68e5·25-s − 1.02e5i·27-s + 3.10e5·29-s − 4.94e5i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.238·5-s + 1.33i·7-s − 0.333·9-s + 0.220i·11-s − 0.00618·13-s + 0.137i·15-s + 0.963·17-s + 1.83i·19-s − 0.771·21-s + 0.304i·23-s − 0.942·25-s − 0.192i·27-s + 0.438·29-s − 0.535i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-1$
Analytic conductor: \(78.2166\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.457296654\)
\(L(\frac12)\) \(\approx\) \(1.457296654\)
\(L(5)\) 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 \( 1 - 46.7iT \)
good5 \( 1 - 149.T + 3.90e5T^{2} \)
7 \( 1 - 3.20e3iT - 5.76e6T^{2} \)
11 \( 1 - 3.23e3iT - 2.14e8T^{2} \)
13 \( 1 + 176.T + 8.15e8T^{2} \)
17 \( 1 - 8.04e4T + 6.97e9T^{2} \)
19 \( 1 - 2.39e5iT - 1.69e10T^{2} \)
23 \( 1 - 8.53e4iT - 7.83e10T^{2} \)
29 \( 1 - 3.10e5T + 5.00e11T^{2} \)
31 \( 1 + 4.94e5iT - 8.52e11T^{2} \)
37 \( 1 - 8.55e5T + 3.51e12T^{2} \)
41 \( 1 + 1.41e6T + 7.98e12T^{2} \)
43 \( 1 - 2.15e6iT - 1.16e13T^{2} \)
47 \( 1 + 1.10e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.47e7T + 6.22e13T^{2} \)
59 \( 1 - 1.62e7iT - 1.46e14T^{2} \)
61 \( 1 + 9.65e6T + 1.91e14T^{2} \)
67 \( 1 - 2.25e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.04e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.01e7T + 8.06e14T^{2} \)
79 \( 1 + 4.96e7iT - 1.51e15T^{2} \)
83 \( 1 + 4.44e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.11e8T + 3.93e15T^{2} \)
97 \( 1 - 3.96e6T + 7.83e15T^{2} \)
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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

−11.79210498736156902238072281864, −10.31779293911077793850648987679, −9.675872307128389903324320288895, −8.657289876618764069965169514690, −7.70389141813213256569150306154, −5.98226752805559162119398486215, −5.50072546949588278879159176473, −4.04755181284098766836973355695, −2.81427537888488039488729688397, −1.59079465831227328734475564940, 0.34776173807277469841245632113, 1.24874544429220755576545746284, 2.73499754409042352389481315307, 4.03366039442386837340705948442, 5.32005427893811203891815303254, 6.65331631619584571857750740137, 7.35982923411831833241512966206, 8.411693774548415863597285328595, 9.660107772692137153962546178274, 10.61060654342331865837115185403

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