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Properties

Label 2-192-16.3-c8-0-11
Degree $2$
Conductor $192$
Sign $0.449 - 0.893i$
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  + (33.0 − 33.0i)3-s + (260. − 260. i)5-s + 368.·7-s − 2.18e3i·9-s + (1.70e4 + 1.70e4i)11-s + (3.22e4 + 3.22e4i)13-s − 1.72e4i·15-s − 1.20e5·17-s + (−5.00e3 + 5.00e3i)19-s + (1.21e4 − 1.21e4i)21-s − 4.28e5·23-s + 2.55e5i·25-s + (−7.23e4 − 7.23e4i)27-s + (6.69e5 + 6.69e5i)29-s + 3.56e4i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.416 − 0.416i)5-s + 0.153·7-s − 0.333i·9-s + (1.16 + 1.16i)11-s + (1.12 + 1.12i)13-s − 0.339i·15-s − 1.44·17-s + (−0.0383 + 0.0383i)19-s + (0.0626 − 0.0626i)21-s − 1.53·23-s + 0.653i·25-s + (−0.136 − 0.136i)27-s + (0.946 + 0.946i)29-s + 0.0385i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(78.2166\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :4),\ 0.449 - 0.893i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.436188904\)
\(L(\frac12)\) \(\approx\) \(2.436188904\)
\(L(5)\) 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 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (-260. + 260. i)T - 3.90e5iT^{2} \)
7 \( 1 - 368.T + 5.76e6T^{2} \)
11 \( 1 + (-1.70e4 - 1.70e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-3.22e4 - 3.22e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 1.20e5T + 6.97e9T^{2} \)
19 \( 1 + (5.00e3 - 5.00e3i)T - 1.69e10iT^{2} \)
23 \( 1 + 4.28e5T + 7.83e10T^{2} \)
29 \( 1 + (-6.69e5 - 6.69e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 3.56e4iT - 8.52e11T^{2} \)
37 \( 1 + (1.95e6 - 1.95e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.75e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.02e6 + 2.02e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 5.13e6iT - 2.38e13T^{2} \)
53 \( 1 + (2.06e6 - 2.06e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (-9.48e6 - 9.48e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (-8.73e6 - 8.73e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.48e7 + 1.48e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 2.71e7T + 6.45e14T^{2} \)
73 \( 1 + 7.57e6iT - 8.06e14T^{2} \)
79 \( 1 + 5.69e6iT - 1.51e15T^{2} \)
83 \( 1 + (7.10e6 - 7.10e6i)T - 2.25e15iT^{2} \)
89 \( 1 - 9.01e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.12e8T + 7.83e15T^{2} \)
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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

−11.46257419338123092124461152202, −10.05494827298080702616118345556, −9.060261141696255560948821283481, −8.484450185001195171336851931996, −6.93072053864486131199605955412, −6.37515051324278592376555846212, −4.72324937299229040828336716657, −3.77057093828644685948150132979, −1.95214241441036812577824383800, −1.42201535137964928102531983889, 0.51209584122039237814187379325, 2.00861176497509495342461399596, 3.29778737220835758306032026115, 4.22832693710513312195167812017, 5.85118108117587557983963549477, 6.52529898752731448148294218434, 8.196498086358660004743419297827, 8.752083417591189060497159145374, 9.955617561708374712394210090762, 10.83770594837195187095581171869

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