# Properties

 Label 2-384-24.5-c6-0-64 Degree $2$ Conductor $384$ Sign $0.163 + 0.986i$ 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 + (4.41 + 26.6i)3-s − 127.·5-s + 467.·7-s + (−689. + 235. i)9-s − 554.·11-s + 357. i·13-s + (−561. − 3.38e3i)15-s + 4.49e3i·17-s + 3.24e3i·19-s + (2.06e3 + 1.24e4i)21-s − 1.74e4i·23-s + 511.·25-s + (−9.31e3 − 1.73e4i)27-s − 1.74e4·29-s + 8.03e3·31-s + ⋯
 L(s)  = 1 + (0.163 + 0.986i)3-s − 1.01·5-s + 1.36·7-s + (−0.946 + 0.322i)9-s − 0.416·11-s + 0.162i·13-s + (−0.166 − 1.00i)15-s + 0.915i·17-s + 0.472i·19-s + (0.223 + 1.34i)21-s − 1.43i·23-s + 0.0327·25-s + (−0.473 − 0.880i)27-s − 0.714·29-s + 0.269·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.163 + 0.986i$ 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: $$(2,\ 384,\ (\ :3),\ 0.163 + 0.986i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.4680344536$$ $$L(\frac12)$$ $$\approx$$ $$0.4680344536$$ $$L(4)$$ 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 + (-4.41 - 26.6i)T$$
good5 $$1 + 127.T + 1.56e4T^{2}$$
7 $$1 - 467.T + 1.17e5T^{2}$$
11 $$1 + 554.T + 1.77e6T^{2}$$
13 $$1 - 357. iT - 4.82e6T^{2}$$
17 $$1 - 4.49e3iT - 2.41e7T^{2}$$
19 $$1 - 3.24e3iT - 4.70e7T^{2}$$
23 $$1 + 1.74e4iT - 1.48e8T^{2}$$
29 $$1 + 1.74e4T + 5.94e8T^{2}$$
31 $$1 - 8.03e3T + 8.87e8T^{2}$$
37 $$1 - 7.94e4iT - 2.56e9T^{2}$$
41 $$1 + 6.18e4iT - 4.75e9T^{2}$$
43 $$1 + 4.54e4iT - 6.32e9T^{2}$$
47 $$1 - 1.59e5iT - 1.07e10T^{2}$$
53 $$1 + 1.90e5T + 2.21e10T^{2}$$
59 $$1 + 3.06e5T + 4.21e10T^{2}$$
61 $$1 + 2.51e5iT - 5.15e10T^{2}$$
67 $$1 + 2.02e5iT - 9.04e10T^{2}$$
71 $$1 + 3.62e5iT - 1.28e11T^{2}$$
73 $$1 + 3.49e5T + 1.51e11T^{2}$$
79 $$1 - 7.18e5T + 2.43e11T^{2}$$
83 $$1 - 1.38e5T + 3.26e11T^{2}$$
89 $$1 + 1.22e6iT - 4.96e11T^{2}$$
97 $$1 - 1.15e6T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$