Properties

Label 2-192-3.2-c8-0-48
Degree $2$
Conductor $192$
Sign $0.555 + 0.831i$
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  + (45 + 67.3i)3-s − 224. i·5-s + 1.75e3·7-s + (−2.51e3 + 6.06e3i)9-s − 6.95e3i·11-s − 2.57e4·13-s + (1.51e4 − 1.01e4i)15-s − 7.48e4i·17-s + 1.89e4·19-s + (7.87e4 + 1.17e5i)21-s − 4.70e5i·23-s + 3.40e5·25-s + (−5.21e5 + 1.03e5i)27-s + 4.60e5i·29-s + 3.51e5·31-s + ⋯
L(s)  = 1  + (0.555 + 0.831i)3-s − 0.359i·5-s + 0.728·7-s + (−0.382 + 0.923i)9-s − 0.475i·11-s − 0.900·13-s + (0.298 − 0.199i)15-s − 0.896i·17-s + 0.145·19-s + (0.404 + 0.606i)21-s − 1.68i·23-s + 0.870·25-s + (−0.980 + 0.195i)27-s + 0.651i·29-s + 0.380·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(78.2166\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :4),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.079710934\)
\(L(\frac12)\) \(\approx\) \(2.079710934\)
\(L(5)\) 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 \( 1 + (-45 - 67.3i)T \)
good5 \( 1 + 224. iT - 3.90e5T^{2} \)
7 \( 1 - 1.75e3T + 5.76e6T^{2} \)
11 \( 1 + 6.95e3iT - 2.14e8T^{2} \)
13 \( 1 + 2.57e4T + 8.15e8T^{2} \)
17 \( 1 + 7.48e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.89e4T + 1.69e10T^{2} \)
23 \( 1 + 4.70e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.60e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.51e5T + 8.52e11T^{2} \)
37 \( 1 + 1.33e6T + 3.51e12T^{2} \)
41 \( 1 + 1.87e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.52e6T + 1.16e13T^{2} \)
47 \( 1 + 4.08e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.60e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.37e7iT - 1.46e14T^{2} \)
61 \( 1 + 7.53e5T + 1.91e14T^{2} \)
67 \( 1 - 2.26e6T + 4.06e14T^{2} \)
71 \( 1 - 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.76e7T + 8.06e14T^{2} \)
79 \( 1 - 2.29e7T + 1.51e15T^{2} \)
83 \( 1 - 4.63e7iT - 2.25e15T^{2} \)
89 \( 1 + 7.26e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.47e8T + 7.83e15T^{2} \)
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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

−10.78027315923164073354451235389, −9.900250127812971928510953078256, −8.821862544453052927987017647861, −8.197969956219001378238944684681, −6.92565056130325057527748665359, −5.17020925726390311216297228226, −4.68283541085945721547247226635, −3.24743228928982274160706143898, −2.11468252608290202986388957272, −0.44415850645929573771848680659, 1.24576636162076006430012520972, 2.20009132927021261601765470668, 3.41550919490135299273803049760, 4.87415422025947824325835258399, 6.23492168679873949409838594243, 7.34540796917537938650444444446, 7.966808776946808401961197392347, 9.115432885666015484659206527134, 10.16087794975541997920810609144, 11.41374147273439395643806029111

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