# Properties

 Label 2-384-8.3-c6-0-39 Degree $2$ Conductor $384$ Sign $i$ Analytic cond. $88.3407$ Root an. cond. $9.39897$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 15.5·3-s + 20i·5-s − 529. i·7-s + 243·9-s + 435.·11-s − 341. i·13-s + 311. i·15-s + 7.68e3·17-s − 4.30e3·19-s − 8.25e3i·21-s + 3.17e3i·23-s + 1.52e4·25-s + 3.78e3·27-s + 1.94e4i·29-s − 1.52e4i·31-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.160i·5-s − 1.54i·7-s + 0.333·9-s + 0.327·11-s − 0.155i·13-s + 0.0923i·15-s + 1.56·17-s − 0.626·19-s − 0.891i·21-s + 0.260i·23-s + 0.974·25-s + 0.192·27-s + 0.795i·29-s − 0.513i·31-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $i$ Analytic conductor: $$88.3407$$ Root analytic conductor: $$9.39897$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{384} (319, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :3),\ i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$2.613303952$$ $$L(\frac12)$$ $$\approx$$ $$2.613303952$$ $$L(4)$$ 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 - 15.5T$$
good5 $$1 - 20iT - 1.56e4T^{2}$$
7 $$1 + 529. iT - 1.17e5T^{2}$$
11 $$1 - 435.T + 1.77e6T^{2}$$
13 $$1 + 341. iT - 4.82e6T^{2}$$
17 $$1 - 7.68e3T + 2.41e7T^{2}$$
19 $$1 + 4.30e3T + 4.70e7T^{2}$$
23 $$1 - 3.17e3iT - 1.48e8T^{2}$$
29 $$1 - 1.94e4iT - 5.94e8T^{2}$$
31 $$1 + 1.52e4iT - 8.87e8T^{2}$$
37 $$1 + 6.19e4iT - 2.56e9T^{2}$$
41 $$1 + 3.37e4T + 4.75e9T^{2}$$
43 $$1 + 9.93e4T + 6.32e9T^{2}$$
47 $$1 - 1.77e4iT - 1.07e10T^{2}$$
53 $$1 + 2.24e5iT - 2.21e10T^{2}$$
59 $$1 - 1.99e5T + 4.21e10T^{2}$$
61 $$1 - 4.56e4iT - 5.15e10T^{2}$$
67 $$1 + 4.96e5T + 9.04e10T^{2}$$
71 $$1 + 4.52e5iT - 1.28e11T^{2}$$
73 $$1 - 3.94e5T + 1.51e11T^{2}$$
79 $$1 + 5.71e5iT - 2.43e11T^{2}$$
83 $$1 - 3.24e5T + 3.26e11T^{2}$$
89 $$1 - 7.58e5T + 4.96e11T^{2}$$
97 $$1 - 2.50e4T + 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

−10.22811087655399796309885771055, −9.242199767428124816642404893397, −8.107538850291593506510739822745, −7.37013048196495235515487507469, −6.55574404013286361219000797159, −5.10493363208262211575299654447, −3.91506123677362789707811725441, −3.21890734905149019689509879670, −1.62077901557472089359438931819, −0.56523714785408119841472428900, 1.26103648517830252407875531868, 2.44132941208016368323196081874, 3.36504295382554983447618911512, 4.77015767518912019998477557845, 5.75043372055172399024420265517, 6.75632322525468159651359487844, 8.089257996738334909749628182349, 8.668428576396202020634971631638, 9.501718782530458718029127619327, 10.37475009807122710433338073589