# Properties

 Degree $2$ Conductor $384$ Sign $-0.732 - 0.680i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.22 − 1.22i)3-s + (5.24 + 5.24i)5-s − 5.32·7-s + 2.99i·9-s + (−12.2 + 12.2i)11-s + (5.73 − 5.73i)13-s − 12.8i·15-s − 23.3·17-s + (−11.7 − 11.7i)19-s + (6.52 + 6.52i)21-s + 5.80·23-s + 29.9i·25-s + (3.67 − 3.67i)27-s + (−18.3 + 18.3i)29-s + 16.9i·31-s + ⋯
 L(s)  = 1 + (−0.408 − 0.408i)3-s + (1.04 + 1.04i)5-s − 0.761·7-s + 0.333i·9-s + (−1.11 + 1.11i)11-s + (0.441 − 0.441i)13-s − 0.856i·15-s − 1.37·17-s + (−0.618 − 0.618i)19-s + (0.310 + 0.310i)21-s + 0.252·23-s + 1.19i·25-s + (0.136 − 0.136i)27-s + (−0.634 + 0.634i)29-s + 0.545i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $-0.732 - 0.680i$ Motivic weight: $$2$$ Character: $\chi_{384} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1),\ -0.732 - 0.680i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.262002 + 0.666871i$$ $$L(\frac12)$$ $$\approx$$ $$0.262002 + 0.666871i$$ $$L(2)$$ 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 + (1.22 + 1.22i)T$$
good5 $$1 + (-5.24 - 5.24i)T + 25iT^{2}$$
7 $$1 + 5.32T + 49T^{2}$$
11 $$1 + (12.2 - 12.2i)T - 121iT^{2}$$
13 $$1 + (-5.73 + 5.73i)T - 169iT^{2}$$
17 $$1 + 23.3T + 289T^{2}$$
19 $$1 + (11.7 + 11.7i)T + 361iT^{2}$$
23 $$1 - 5.80T + 529T^{2}$$
29 $$1 + (18.3 - 18.3i)T - 841iT^{2}$$
31 $$1 - 16.9iT - 961T^{2}$$
37 $$1 + (15.3 + 15.3i)T + 1.36e3iT^{2}$$
41 $$1 - 29.2iT - 1.68e3T^{2}$$
43 $$1 + (33.4 - 33.4i)T - 1.84e3iT^{2}$$
47 $$1 - 18.2iT - 2.20e3T^{2}$$
53 $$1 + (-66.9 - 66.9i)T + 2.80e3iT^{2}$$
59 $$1 + (-27.1 + 27.1i)T - 3.48e3iT^{2}$$
61 $$1 + (65.2 - 65.2i)T - 3.72e3iT^{2}$$
67 $$1 + (-37.6 - 37.6i)T + 4.48e3iT^{2}$$
71 $$1 - 42.6T + 5.04e3T^{2}$$
73 $$1 + 106. iT - 5.32e3T^{2}$$
79 $$1 - 21.2iT - 6.24e3T^{2}$$
83 $$1 + (24.1 + 24.1i)T + 6.88e3iT^{2}$$
89 $$1 + 52.8iT - 7.92e3T^{2}$$
97 $$1 + 21.0T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$