↑

Properties

Label 2-192-16.11-c8-0-15
Degree $2$
Conductor $192$
Sign $0.417 - 0.908i$
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 + (625. + 625. i)5-s − 873.·7-s + 2.18e3i·9-s + (7.49e3 − 7.49e3i)11-s + (−1.76e4 + 1.76e4i)13-s − 4.14e4i·15-s + 1.20e5·17-s + (2.06e4 + 2.06e4i)19-s + (2.88e4 + 2.88e4i)21-s + 4.99e3·23-s + 3.93e5i·25-s + (7.23e4 − 7.23e4i)27-s + (5.91e5 − 5.91e5i)29-s + 4.99e5i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.00 + 1.00i)5-s − 0.363·7-s + 0.333i·9-s + (0.511 − 0.511i)11-s + (−0.616 + 0.616i)13-s − 0.817i·15-s + 1.43·17-s + (0.158 + 0.158i)19-s + (0.148 + 0.148i)21-s + 0.0178·23-s + 1.00i·25-s + (0.136 − 0.136i)27-s + (0.836 − 0.836i)29-s + 0.540i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.417 - 0.908i$
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: \((2,\ 192,\ (\ :4),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.062031039\)
\(L(\frac12)\) \(\approx\) \(2.062031039\)
\(L(5)\) 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 \( 1 + (-625. - 625. i)T + 3.90e5iT^{2} \)
7 \( 1 + 873.T + 5.76e6T^{2} \)
11 \( 1 + (-7.49e3 + 7.49e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (1.76e4 - 1.76e4i)T - 8.15e8iT^{2} \)
17 \( 1 - 1.20e5T + 6.97e9T^{2} \)
19 \( 1 + (-2.06e4 - 2.06e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 4.99e3T + 7.83e10T^{2} \)
29 \( 1 + (-5.91e5 + 5.91e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 4.99e5iT - 8.52e11T^{2} \)
37 \( 1 + (5.48e5 + 5.48e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.47e6iT - 7.98e12T^{2} \)
43 \( 1 + (1.86e6 - 1.86e6i)T - 1.16e13iT^{2} \)
47 \( 1 + 3.07e6iT - 2.38e13T^{2} \)
53 \( 1 + (8.63e6 + 8.63e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (4.64e6 - 4.64e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-1.74e4 + 1.74e4i)T - 1.91e14iT^{2} \)
67 \( 1 + (-2.63e7 - 2.63e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 4.89e7T + 6.45e14T^{2} \)
73 \( 1 - 5.31e7iT - 8.06e14T^{2} \)
79 \( 1 - 6.50e6iT - 1.51e15T^{2} \)
83 \( 1 + (-5.61e7 - 5.61e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 8.95e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.32e8T + 7.83e15T^{2} \)
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

−11.27710262971811470012558529231, −10.11872767366518843914241845637, −9.614501032788659198204906155416, −8.080260032787372683301437563345, −6.81679879296629897039528024953, −6.27245361709287078417764122603, −5.19589222495476046740099423693, −3.43934581157863474500165977602, −2.32245659890195068177484978564, −1.08167646778173446180715594813, 0.56854734654176399656303539426, 1.67313319257612581517229240575, 3.26367615575215928107245025994, 4.78523014619701558238222154377, 5.45664213578114973517350113441, 6.52398738250600922090579350389, 7.919286746072124665193033654947, 9.274295095119732636546537079071, 9.731301346256223731201446621736, 10.65266111530635187113369980576

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