# Properties

 Label 2-384-48.5-c2-0-13 Degree $2$ Conductor $384$ Sign $0.961 - 0.274i$ Analytic cond. $10.4632$ Root an. cond. $3.23469$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.17 + 2.06i)3-s + (−3.17 + 3.17i)5-s − 6.03i·7-s + (0.485 − 8.98i)9-s + (13.0 − 13.0i)11-s + (−6.39 + 6.39i)13-s + (0.363 − 13.4i)15-s + 4.39i·17-s + (−3.21 + 3.21i)19-s + (12.4 + 13.1i)21-s + 34.0·23-s + 4.78i·25-s + (17.4 + 20.5i)27-s + (27.9 + 27.9i)29-s + 7.90·31-s + ⋯
 L(s)  = 1 + (−0.725 + 0.687i)3-s + (−0.635 + 0.635i)5-s − 0.862i·7-s + (0.0539 − 0.998i)9-s + (1.18 − 1.18i)11-s + (−0.491 + 0.491i)13-s + (0.0242 − 0.898i)15-s + 0.258i·17-s + (−0.168 + 0.168i)19-s + (0.593 + 0.626i)21-s + 1.47·23-s + 0.191i·25-s + (0.647 + 0.761i)27-s + (0.964 + 0.964i)29-s + 0.255·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.961 - 0.274i$ Analytic conductor: $$10.4632$$ Root analytic conductor: $$3.23469$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{384} (353, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1),\ 0.961 - 0.274i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.15547 + 0.161704i$$ $$L(\frac12)$$ $$\approx$$ $$1.15547 + 0.161704i$$ $$L(2)$$ 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 + (2.17 - 2.06i)T$$
good5 $$1 + (3.17 - 3.17i)T - 25iT^{2}$$
7 $$1 + 6.03iT - 49T^{2}$$
11 $$1 + (-13.0 + 13.0i)T - 121iT^{2}$$
13 $$1 + (6.39 - 6.39i)T - 169iT^{2}$$
17 $$1 - 4.39iT - 289T^{2}$$
19 $$1 + (3.21 - 3.21i)T - 361iT^{2}$$
23 $$1 - 34.0T + 529T^{2}$$
29 $$1 + (-27.9 - 27.9i)T + 841iT^{2}$$
31 $$1 - 7.90T + 961T^{2}$$
37 $$1 + (20.0 + 20.0i)T + 1.36e3iT^{2}$$
41 $$1 - 45.1T + 1.68e3T^{2}$$
43 $$1 + (36.0 + 36.0i)T + 1.84e3iT^{2}$$
47 $$1 - 5.08iT - 2.20e3T^{2}$$
53 $$1 + (-20.7 + 20.7i)T - 2.80e3iT^{2}$$
59 $$1 + (-39.0 + 39.0i)T - 3.48e3iT^{2}$$
61 $$1 + (-49.8 + 49.8i)T - 3.72e3iT^{2}$$
67 $$1 + (-44.9 + 44.9i)T - 4.48e3iT^{2}$$
71 $$1 - 46.6T + 5.04e3T^{2}$$
73 $$1 - 97.3iT - 5.32e3T^{2}$$
79 $$1 - 40.1T + 6.24e3T^{2}$$
83 $$1 + (-35.5 - 35.5i)T + 6.88e3iT^{2}$$
89 $$1 + 69.6T + 7.92e3T^{2}$$
97 $$1 - 61.0T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$