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Properties

Label 2-192-48.5-c2-0-4
Degree $2$
Conductor $192$
Sign $-0.358 - 0.933i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.77 + 1.13i)3-s + (−6.28 + 6.28i)5-s − 1.64i·7-s + (6.43 + 6.29i)9-s + (−4.75 + 4.75i)11-s + (−9.35 + 9.35i)13-s + (−24.5 + 10.3i)15-s + 11.4i·17-s + (8.58 − 8.58i)19-s + (1.86 − 4.57i)21-s + 16.2·23-s − 54.0i·25-s + (10.7 + 24.7i)27-s + (10.7 + 10.7i)29-s − 6.35·31-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)3-s + (−1.25 + 1.25i)5-s − 0.235i·7-s + (0.715 + 0.699i)9-s + (−0.432 + 0.432i)11-s + (−0.719 + 0.719i)13-s + (−1.63 + 0.689i)15-s + 0.675i·17-s + (0.451 − 0.451i)19-s + (0.0887 − 0.217i)21-s + 0.706·23-s − 2.16i·25-s + (0.398 + 0.917i)27-s + (0.370 + 0.370i)29-s − 0.204·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.809989 + 1.17918i\)
\(L(\frac12)\) \(\approx\) \(0.809989 + 1.17918i\)
\(L(2)\) 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 \( 1 + (-2.77 - 1.13i)T \)
good5 \( 1 + (6.28 - 6.28i)T - 25iT^{2} \)
7 \( 1 + 1.64iT - 49T^{2} \)
11 \( 1 + (4.75 - 4.75i)T - 121iT^{2} \)
13 \( 1 + (9.35 - 9.35i)T - 169iT^{2} \)
17 \( 1 - 11.4iT - 289T^{2} \)
19 \( 1 + (-8.58 + 8.58i)T - 361iT^{2} \)
23 \( 1 - 16.2T + 529T^{2} \)
29 \( 1 + (-10.7 - 10.7i)T + 841iT^{2} \)
31 \( 1 + 6.35T + 961T^{2} \)
37 \( 1 + (-27.2 - 27.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.98T + 1.68e3T^{2} \)
43 \( 1 + (-19.4 - 19.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (4.00 - 4.00i)T - 2.80e3iT^{2} \)
59 \( 1 + (27.9 - 27.9i)T - 3.48e3iT^{2} \)
61 \( 1 + (-39.2 + 39.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-68.6 + 68.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 40.6T + 5.04e3T^{2} \)
73 \( 1 - 59.0iT - 5.32e3T^{2} \)
79 \( 1 + 17.3T + 6.24e3T^{2} \)
83 \( 1 + (75.1 + 75.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−12.59632046666483497702347177804, −11.46326081684758548060338438502, −10.61852317315182258386916347078, −9.741998713477262614370464126292, −8.444374866158069072160267669451, −7.46647093766185320140497559318, −6.88249771202581406932953121639, −4.68861510215831280088986561437, −3.64130813587629485274942611988, −2.55059706610198536308316647099, 0.76972115015039227886187311764, 2.88190735855970880694419593674, 4.19622509605699924584103702617, 5.39514927328910254632705294268, 7.32929162337966567471078079600, 7.971776067079739265191179480164, 8.785978708960119510935578083426, 9.697287748425241268706501755949, 11.25997191035437021427808419970, 12.35200433688280270891122701349

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