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Properties

Label 2-192-16.3-c8-0-15
Degree $2$
Conductor $192$
Sign $0.949 - 0.314i$
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 + (214. − 214. i)5-s + 321.·7-s − 2.18e3i·9-s + (1.34e4 + 1.34e4i)11-s + (−3.43e3 − 3.43e3i)13-s − 1.41e4i·15-s + 6.59e4·17-s + (−1.13e5 + 1.13e5i)19-s + (1.06e4 − 1.06e4i)21-s + 9.91e4·23-s + 2.98e5i·25-s + (−7.23e4 − 7.23e4i)27-s + (7.42e4 + 7.42e4i)29-s + 7.83e5i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.343 − 0.343i)5-s + 0.133·7-s − 0.333i·9-s + (0.918 + 0.918i)11-s + (−0.120 − 0.120i)13-s − 0.280i·15-s + 0.789·17-s + (−0.871 + 0.871i)19-s + (0.0546 − 0.0546i)21-s + 0.354·23-s + 0.764i·25-s + (−0.136 − 0.136i)27-s + (0.104 + 0.104i)29-s + 0.848i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.949 - 0.314i$
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: \((2,\ 192,\ (\ :4),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.849807584\)
\(L(\frac12)\) \(\approx\) \(2.849807584\)
\(L(5)\) 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 \( 1 + (-214. + 214. i)T - 3.90e5iT^{2} \)
7 \( 1 - 321.T + 5.76e6T^{2} \)
11 \( 1 + (-1.34e4 - 1.34e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (3.43e3 + 3.43e3i)T + 8.15e8iT^{2} \)
17 \( 1 - 6.59e4T + 6.97e9T^{2} \)
19 \( 1 + (1.13e5 - 1.13e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 9.91e4T + 7.83e10T^{2} \)
29 \( 1 + (-7.42e4 - 7.42e4i)T + 5.00e11iT^{2} \)
31 \( 1 - 7.83e5iT - 8.52e11T^{2} \)
37 \( 1 + (-3.85e5 + 3.85e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.10e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.77e6 - 2.77e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 6.61e6iT - 2.38e13T^{2} \)
53 \( 1 + (-8.73e5 + 8.73e5i)T - 6.22e13iT^{2} \)
59 \( 1 + (-1.79e6 - 1.79e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.03e7 + 1.03e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.57e7 + 1.57e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 4.51e7T + 6.45e14T^{2} \)
73 \( 1 - 1.41e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.93e6iT - 1.51e15T^{2} \)
83 \( 1 + (4.07e7 - 4.07e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 3.55e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.43e7T + 7.83e15T^{2} \)
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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

−11.15477269430771252994122102233, −9.903011315506789758823109816052, −9.157808901911472228965985892492, −8.081524727661821066912379801467, −7.08266207707538689665257726983, −6.00201856097679685121538000876, −4.70734902003395064455374249700, −3.46615411071888473999106181749, −1.98015095843310452249645602321, −1.11947530060154369669172060812, 0.69790958162050526319352189079, 2.20917673167449566620403375798, 3.36208738468383050381197834665, 4.48323290732880277294777350523, 5.84457275534740327713776118527, 6.83239523814517639112769968748, 8.174307377711702191868393486605, 9.034018484953220132214672000768, 9.977051590816397260454465987787, 10.95229832746839934558131889765

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