# Properties

 Label 2-384-8.5-c3-0-18 Degree $2$ Conductor $384$ Sign $i$ Analytic cond. $22.6567$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·3-s + 2.82i·5-s + 14.1·7-s − 9·9-s − 20i·11-s − 39.5i·13-s + 8.48·15-s − 34·17-s + 52i·19-s − 42.4i·21-s + 62.2·23-s + 117·25-s + 27i·27-s − 200. i·29-s + 110.·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + 0.252i·5-s + 0.763·7-s − 0.333·9-s − 0.548i·11-s − 0.844i·13-s + 0.146·15-s − 0.485·17-s + 0.627i·19-s − 0.440i·21-s + 0.564·23-s + 0.936·25-s + 0.192i·27-s − 1.28i·29-s + 0.639·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $i$ Analytic conductor: $$22.6567$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{384} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :3/2),\ i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.786716837$$ $$L(\frac12)$$ $$\approx$$ $$1.786716837$$ $$L(\frac{5}{2})$$ 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 + 3iT$$
good5 $$1 - 2.82iT - 125T^{2}$$
7 $$1 - 14.1T + 343T^{2}$$
11 $$1 + 20iT - 1.33e3T^{2}$$
13 $$1 + 39.5iT - 2.19e3T^{2}$$
17 $$1 + 34T + 4.91e3T^{2}$$
19 $$1 - 52iT - 6.85e3T^{2}$$
23 $$1 - 62.2T + 1.21e4T^{2}$$
29 $$1 + 200. iT - 2.43e4T^{2}$$
31 $$1 - 110.T + 2.97e4T^{2}$$
37 $$1 + 271. iT - 5.06e4T^{2}$$
41 $$1 - 26T + 6.89e4T^{2}$$
43 $$1 + 252iT - 7.95e4T^{2}$$
47 $$1 + 345.T + 1.03e5T^{2}$$
53 $$1 + 681. iT - 1.48e5T^{2}$$
59 $$1 + 364iT - 2.05e5T^{2}$$
61 $$1 + 735. iT - 2.26e5T^{2}$$
67 $$1 - 628iT - 3.00e5T^{2}$$
71 $$1 + 333.T + 3.57e5T^{2}$$
73 $$1 + 338T + 3.89e5T^{2}$$
79 $$1 - 789.T + 4.93e5T^{2}$$
83 $$1 - 1.03e3iT - 5.71e5T^{2}$$
89 $$1 + 234T + 7.04e5T^{2}$$
97 $$1 + 178T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$