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Properties

Label 2-192-16.3-c8-0-31
Degree $2$
Conductor $192$
Sign $-0.999 - 0.0156i$
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 + (691. − 691. i)5-s + 1.74e3·7-s − 2.18e3i·9-s + (−4.44e3 − 4.44e3i)11-s + (−2.60e4 − 2.60e4i)13-s − 4.57e4i·15-s − 3.35e4·17-s + (−1.56e5 + 1.56e5i)19-s + (5.77e4 − 5.77e4i)21-s − 4.53e5·23-s − 5.65e5i·25-s + (−7.23e4 − 7.23e4i)27-s + (−3.89e5 − 3.89e5i)29-s − 5.54e5i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.10 − 1.10i)5-s + 0.727·7-s − 0.333i·9-s + (−0.303 − 0.303i)11-s + (−0.912 − 0.912i)13-s − 0.903i·15-s − 0.401·17-s + (−1.20 + 1.20i)19-s + (0.296 − 0.296i)21-s − 1.62·23-s − 1.44i·25-s + (−0.136 − 0.136i)27-s + (−0.551 − 0.551i)29-s − 0.600i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0156i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.999 - 0.0156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.999 - 0.0156i$
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: \((2,\ 192,\ (\ :4),\ -0.999 - 0.0156i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.636433290\)
\(L(\frac12)\) \(\approx\) \(1.636433290\)
\(L(5)\) 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 \( 1 + (-691. + 691. i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.74e3T + 5.76e6T^{2} \)
11 \( 1 + (4.44e3 + 4.44e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (2.60e4 + 2.60e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 3.35e4T + 6.97e9T^{2} \)
19 \( 1 + (1.56e5 - 1.56e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 4.53e5T + 7.83e10T^{2} \)
29 \( 1 + (3.89e5 + 3.89e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 5.54e5iT - 8.52e11T^{2} \)
37 \( 1 + (-4.85e4 + 4.85e4i)T - 3.51e12iT^{2} \)
41 \( 1 - 4.43e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.65e6 - 2.65e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 5.54e6iT - 2.38e13T^{2} \)
53 \( 1 + (-3.88e6 + 3.88e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (3.06e6 + 3.06e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (-1.78e7 - 1.78e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (-2.35e7 + 2.35e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 9.17e6T + 6.45e14T^{2} \)
73 \( 1 - 1.49e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.39e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.29e7 - 1.29e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 7.81e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.80e7T + 7.83e15T^{2} \)
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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

−10.30210949167316593511717882786, −9.582480895845738513550224235211, −8.344186740855366593708229751271, −7.911327746829711372369308089166, −6.18191593803240281928198466296, −5.37883692171886281176120787126, −4.21192921838942027366740825675, −2.36182261101880817617851695692, −1.63437273950230226274747866471, −0.30192022921749236382689654991, 2.04267998239496490560400666681, 2.41057196548407105247515611739, 4.09764388397548432401388067339, 5.21504541924128657574545031537, 6.51361089818113833873714518852, 7.37769246292873375320289534014, 8.745571797695591075281682576121, 9.669543035537950606954617806113, 10.51564147840617889772504675558, 11.22758289375708969826262806369

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