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Properties

Label 2-192-48.5-c2-0-6
Degree $2$
Conductor $192$
Sign $0.884 + 0.466i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
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  + (−2.75 − 1.18i)3-s + (0.00985 − 0.00985i)5-s + 6.42i·7-s + (6.19 + 6.53i)9-s + (9.07 − 9.07i)11-s + (12.6 − 12.6i)13-s + (−0.0388 + 0.0154i)15-s − 19.0i·17-s + (2.07 − 2.07i)19-s + (7.61 − 17.7i)21-s + 19.5·23-s + 24.9i·25-s + (−9.32 − 25.3i)27-s + (11.1 + 11.1i)29-s + 59.9·31-s + ⋯
L(s)  = 1  + (−0.918 − 0.395i)3-s + (0.00197 − 0.00197i)5-s + 0.917i·7-s + (0.687 + 0.725i)9-s + (0.824 − 0.824i)11-s + (0.969 − 0.969i)13-s + (−0.00259 + 0.00103i)15-s − 1.11i·17-s + (0.109 − 0.109i)19-s + (0.362 − 0.842i)21-s + 0.850·23-s + 0.999i·25-s + (−0.345 − 0.938i)27-s + (0.385 + 0.385i)29-s + 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17740 - 0.291365i\)
\(L(\frac12)\) \(\approx\) \(1.17740 - 0.291365i\)
\(L(2)\) 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 \( 1 + (2.75 + 1.18i)T \)
good5 \( 1 + (-0.00985 + 0.00985i)T - 25iT^{2} \)
7 \( 1 - 6.42iT - 49T^{2} \)
11 \( 1 + (-9.07 + 9.07i)T - 121iT^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (-2.07 + 2.07i)T - 361iT^{2} \)
23 \( 1 - 19.5T + 529T^{2} \)
29 \( 1 + (-11.1 - 11.1i)T + 841iT^{2} \)
31 \( 1 - 59.9T + 961T^{2} \)
37 \( 1 + (-9.32 - 9.32i)T + 1.36e3iT^{2} \)
41 \( 1 + 47.2T + 1.68e3T^{2} \)
43 \( 1 + (24.1 + 24.1i)T + 1.84e3iT^{2} \)
47 \( 1 - 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (20.6 - 20.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (60.3 - 60.3i)T - 3.48e3iT^{2} \)
61 \( 1 + (-48.0 + 48.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-23.7 + 23.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 13.5T + 5.04e3T^{2} \)
73 \( 1 + 31.4iT - 5.32e3T^{2} \)
79 \( 1 + 47.4T + 6.24e3T^{2} \)
83 \( 1 + (70.3 + 70.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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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

−11.98897123985067740498638540829, −11.47787662961969155754159336220, −10.50001729895025364562352015292, −9.178727913881203698945433676094, −8.198920646523535542436520754639, −6.81999045552760475007060121308, −5.90512932109711503324449370780, −4.98087726697034053161385118432, −3.10012722102016897548852901680, −1.03910337329935522637198436731, 1.27163178558634973126273378519, 3.89746200756965603984177466549, 4.61405747416604631236006600238, 6.30676970004766187481703191437, 6.83976078532060958813806891461, 8.406143858735649486438775665098, 9.700175257860252405765154757423, 10.43297753011766089210217698596, 11.40117904330760136570724317775, 12.17151848407785711934777397673

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