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Properties

Label 2-192-12.11-c3-0-5
Degree $2$
Conductor $192$
Sign $-0.995 - 0.0957i$
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  + (−0.497 + 5.17i)3-s + 18.4i·5-s + 17.1i·7-s + (−26.5 − 5.14i)9-s + 48.1·11-s − 33.7·13-s + (−95.4 − 9.18i)15-s − 84.4i·17-s + 28.7i·19-s + (−88.7 − 8.54i)21-s + 96.4·23-s − 215.·25-s + (39.8 − 134. i)27-s + 141. i·29-s − 51.8i·31-s + ⋯
L(s)  = 1  + (−0.0957 + 0.995i)3-s + 1.65i·5-s + 0.926i·7-s + (−0.981 − 0.190i)9-s + 1.32·11-s − 0.719·13-s + (−1.64 − 0.158i)15-s − 1.20i·17-s + 0.346i·19-s + (−0.922 − 0.0887i)21-s + 0.874·23-s − 1.72·25-s + (0.283 − 0.958i)27-s + 0.904i·29-s − 0.300i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.995 - 0.0957i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.995 - 0.0957i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0664733 + 1.38476i\)
\(L(\frac12)\) \(\approx\) \(0.0664733 + 1.38476i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.497 - 5.17i)T \)
good5 \( 1 - 18.4iT - 125T^{2} \)
7 \( 1 - 17.1iT - 343T^{2} \)
11 \( 1 - 48.1T + 1.33e3T^{2} \)
13 \( 1 + 33.7T + 2.19e3T^{2} \)
17 \( 1 + 84.4iT - 4.91e3T^{2} \)
19 \( 1 - 28.7iT - 6.85e3T^{2} \)
23 \( 1 - 96.4T + 1.21e4T^{2} \)
29 \( 1 - 141. iT - 2.43e4T^{2} \)
31 \( 1 + 51.8iT - 2.97e4T^{2} \)
37 \( 1 + 323.T + 5.06e4T^{2} \)
41 \( 1 - 134. iT - 6.89e4T^{2} \)
43 \( 1 - 114. iT - 7.95e4T^{2} \)
47 \( 1 - 247.T + 1.03e5T^{2} \)
53 \( 1 + 169. iT - 1.48e5T^{2} \)
59 \( 1 + 605.T + 2.05e5T^{2} \)
61 \( 1 - 343.T + 2.26e5T^{2} \)
67 \( 1 - 900. iT - 3.00e5T^{2} \)
71 \( 1 + 331.T + 3.57e5T^{2} \)
73 \( 1 - 777.T + 3.89e5T^{2} \)
79 \( 1 + 587. iT - 4.93e5T^{2} \)
83 \( 1 - 1.46e3T + 5.71e5T^{2} \)
89 \( 1 - 78.6iT - 7.04e5T^{2} \)
97 \( 1 + 62.0T + 9.12e5T^{2} \)
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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

−12.10311861470010139162908440115, −11.49526398682543573125788692091, −10.60861639789396877231413559609, −9.627337373523770129202046209643, −8.896547184094849252761186534043, −7.23841914832806993911633644079, −6.31213388003831339708323009342, −5.08853484978308337713871146430, −3.55168465538431775784330239305, −2.59688116755758121386239114792, 0.63725025819006377035291163148, 1.65866903103142836794338295966, 3.95117118308198870212404717444, 5.14184775369171488007570229500, 6.45030808633858606303624543968, 7.50740205566147301918766849767, 8.565105968052333220955724251953, 9.310084712003503901207920980570, 10.76522247470650153900392972832, 12.05279989156460021804819910512

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