Properties

Label 2-384-24.5-c6-0-12
Degree $2$
Conductor $384$
Sign $-0.830 + 0.557i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−22.4 + 15.0i)3-s − 122.·5-s − 206.·7-s + (275. − 674. i)9-s + 142.·11-s + 3.49e3i·13-s + (2.75e3 − 1.84e3i)15-s − 3.38e3i·17-s + 755. i·19-s + (4.63e3 − 3.10e3i)21-s + 1.69e4i·23-s − 561.·25-s + (3.96e3 + 1.92e4i)27-s + 4.11e4·29-s + 2.29e4·31-s + ⋯
L(s)  = 1  + (−0.830 + 0.557i)3-s − 0.981·5-s − 0.602·7-s + (0.378 − 0.925i)9-s + 0.107·11-s + 1.59i·13-s + (0.815 − 0.547i)15-s − 0.688i·17-s + 0.110i·19-s + (0.500 − 0.335i)21-s + 1.38i·23-s − 0.0359·25-s + (0.201 + 0.979i)27-s + 1.68·29-s + 0.769·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.3591410017\)
\(L(\frac12)\) \(\approx\) \(0.3591410017\)
\(L(4)\) 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 \( 1 + (22.4 - 15.0i)T \)
good5 \( 1 + 122.T + 1.56e4T^{2} \)
7 \( 1 + 206.T + 1.17e5T^{2} \)
11 \( 1 - 142.T + 1.77e6T^{2} \)
13 \( 1 - 3.49e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.38e3iT - 2.41e7T^{2} \)
19 \( 1 - 755. iT - 4.70e7T^{2} \)
23 \( 1 - 1.69e4iT - 1.48e8T^{2} \)
29 \( 1 - 4.11e4T + 5.94e8T^{2} \)
31 \( 1 - 2.29e4T + 8.87e8T^{2} \)
37 \( 1 - 2.68e4iT - 2.56e9T^{2} \)
41 \( 1 - 4.04e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.28e5iT - 6.32e9T^{2} \)
47 \( 1 - 6.06e4iT - 1.07e10T^{2} \)
53 \( 1 + 5.29e3T + 2.21e10T^{2} \)
59 \( 1 - 9.73e4T + 4.21e10T^{2} \)
61 \( 1 + 2.13e5iT - 5.15e10T^{2} \)
67 \( 1 - 3.85e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.56e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.41e5T + 1.51e11T^{2} \)
79 \( 1 - 7.22e5T + 2.43e11T^{2} \)
83 \( 1 - 8.04e3T + 3.26e11T^{2} \)
89 \( 1 - 4.10e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.52e6T + 8.32e11T^{2} \)
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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

−11.11954056553006043457748044997, −9.838963193636460875611879656841, −9.354645605883514014752886035933, −8.072972108702526049985885864711, −6.90394769996818885920821388896, −6.26882544419784481230897982253, −4.86078419124724132974666152213, −4.14551111823940627666689581839, −3.10689056580321899390976604942, −1.18486296577278221393234427374, 0.13915291791865138686664786528, 0.819634484598732597117388441498, 2.56043296900980092392223334353, 3.81579865399070435919516897565, 4.97261900383991431766069848471, 6.06036081770862503792069477421, 6.88192322272359590067204439034, 7.898796061014215831048215624096, 8.540025531670674452377237382652, 10.25693448259502822117917515413

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