↑

Properties

Label 2-192-16.11-c8-0-7
Degree $2$
Conductor $192$
Sign $-0.731 - 0.681i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (33.0 + 33.0i)3-s + (−333. − 333. i)5-s + 1.76e3·7-s + 2.18e3i·9-s + (5.11e3 − 5.11e3i)11-s + (−2.43e4 + 2.43e4i)13-s − 2.20e4i·15-s − 1.32e4·17-s + (7.29e3 + 7.29e3i)19-s + (5.83e4 + 5.83e4i)21-s − 2.90e5·23-s − 1.68e5i·25-s + (−7.23e4 + 7.23e4i)27-s + (6.52e5 − 6.52e5i)29-s + 3.86e5i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.533 − 0.533i)5-s + 0.735·7-s + 0.333i·9-s + (0.349 − 0.349i)11-s + (−0.850 + 0.850i)13-s − 0.435i·15-s − 0.158·17-s + (0.0559 + 0.0559i)19-s + (0.300 + 0.300i)21-s − 1.03·23-s − 0.430i·25-s + (−0.136 + 0.136i)27-s + (0.922 − 0.922i)29-s + 0.418i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(78.2166\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :4),\ -0.731 - 0.681i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.028284287\)
\(L(\frac12)\) \(\approx\) \(1.028284287\)
\(L(5)\) 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 \( 1 + (333. + 333. i)T + 3.90e5iT^{2} \)
7 \( 1 - 1.76e3T + 5.76e6T^{2} \)
11 \( 1 + (-5.11e3 + 5.11e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (2.43e4 - 2.43e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 1.32e4T + 6.97e9T^{2} \)
19 \( 1 + (-7.29e3 - 7.29e3i)T + 1.69e10iT^{2} \)
23 \( 1 + 2.90e5T + 7.83e10T^{2} \)
29 \( 1 + (-6.52e5 + 6.52e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 3.86e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.62e6 - 1.62e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 1.15e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.25e6 - 2.25e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 3.16e6iT - 2.38e13T^{2} \)
53 \( 1 + (-8.51e6 - 8.51e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-1.02e6 + 1.02e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (1.85e7 - 1.85e7i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.09e7 + 1.09e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 2.07e7T + 6.45e14T^{2} \)
73 \( 1 + 1.49e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.87e7iT - 1.51e15T^{2} \)
83 \( 1 + (4.60e7 + 4.60e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 1.96e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.61e7T + 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65418575526873251447983483434, −10.36809216551095476822565430243, −9.365194972467930235230381022302, −8.398054708107413552529428446432, −7.69051897639814994402053612380, −6.25940065470856125569906437951, −4.72123209207537185323155198611, −4.20924928047326900770564255811, −2.65060942467798169275375121184, −1.30812686015220987917868910312, 0.22265800864426719544556746530, 1.71133155906358705737409687127, 2.87582639279150601302754530139, 4.10149696727935520802503197377, 5.38341328601206274839176336860, 6.84402390210527829139128205099, 7.63689919248944764213380143766, 8.437498416640392560193040312771, 9.711208146560947494488568278593, 10.73976112741326032163866137871

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