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Properties

Label 2-192-48.11-c3-0-12
Degree $2$
Conductor $192$
Sign $0.765 - 0.643i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.65 + 4.46i)3-s + (5.27 − 5.27i)5-s + 22.9·7-s + (−12.9 + 23.7i)9-s + (10.6 + 10.6i)11-s + (12.7 − 12.7i)13-s + (37.5 + 9.57i)15-s − 134. i·17-s + (46.9 + 46.9i)19-s + (60.8 + 102. i)21-s + 93.7i·23-s + 69.2i·25-s + (−140. + 5.17i)27-s + (161. + 161. i)29-s + 120. i·31-s + ⋯
L(s)  = 1  + (0.510 + 0.859i)3-s + (0.472 − 0.472i)5-s + 1.23·7-s + (−0.478 + 0.878i)9-s + (0.291 + 0.291i)11-s + (0.271 − 0.271i)13-s + (0.646 + 0.164i)15-s − 1.91i·17-s + (0.566 + 0.566i)19-s + (0.632 + 1.06i)21-s + 0.849i·23-s + 0.554i·25-s + (−0.999 + 0.0369i)27-s + (1.03 + 1.03i)29-s + 0.698i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.765 - 0.643i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.36852 + 0.862904i\)
\(L(\frac12)\) \(\approx\) \(2.36852 + 0.862904i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.65 - 4.46i)T \)
good5 \( 1 + (-5.27 + 5.27i)T - 125iT^{2} \)
7 \( 1 - 22.9T + 343T^{2} \)
11 \( 1 + (-10.6 - 10.6i)T + 1.33e3iT^{2} \)
13 \( 1 + (-12.7 + 12.7i)T - 2.19e3iT^{2} \)
17 \( 1 + 134. iT - 4.91e3T^{2} \)
19 \( 1 + (-46.9 - 46.9i)T + 6.85e3iT^{2} \)
23 \( 1 - 93.7iT - 1.21e4T^{2} \)
29 \( 1 + (-161. - 161. i)T + 2.43e4iT^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-2.42 - 2.42i)T + 5.06e4iT^{2} \)
41 \( 1 + 253.T + 6.89e4T^{2} \)
43 \( 1 + (-135. + 135. i)T - 7.95e4iT^{2} \)
47 \( 1 + 468.T + 1.03e5T^{2} \)
53 \( 1 + (-321. + 321. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-119. - 119. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-310. + 310. i)T - 2.26e5iT^{2} \)
67 \( 1 + (705. + 705. i)T + 3.00e5iT^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + 641. iT - 3.89e5T^{2} \)
79 \( 1 + 1.23e3iT - 4.93e5T^{2} \)
83 \( 1 + (87.3 - 87.3i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−11.97603868661459588567987058369, −11.19353622100833825558721478301, −10.05892678125261417838974001043, −9.222327051950166968193331412696, −8.357489352966400057457100408008, −7.27237491535670927641726981383, −5.31025421825926399699019146650, −4.83518703647825219702467493475, −3.24263263723266374733386012045, −1.57389724753921303164397413688, 1.32896420654670648920016949388, 2.51707466887203530034225173633, 4.18331354022819555462058242115, 5.90140982924325820006497346311, 6.75514779262730053926378109720, 8.101947856273995577819565361171, 8.575664819613124403563113992772, 10.02432927440020126465051852395, 11.13336816736301075885134672969, 11.95987544302566858156498712219

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