# 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 $0$

# Related objects

## 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