# Properties

 Label 2-384-24.11-c3-0-41 Degree $2$ Conductor $384$ Sign $0.290 + 0.956i$ 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 + (−2.44 + 4.58i)3-s + 17.2·5-s + 0.269i·7-s + (−15 − 22.4i)9-s − 41.1i·11-s − 64.6i·13-s + (−42.3 + 79.1i)15-s − 63.4i·17-s − 154.·19-s + (−1.23 − 0.660i)21-s − 43.3·23-s + 173.·25-s + (139. − 13.7i)27-s − 240.·29-s − 79.1i·31-s + ⋯
 L(s)  = 1 + (−0.471 + 0.881i)3-s + 1.54·5-s + 0.0145i·7-s + (−0.555 − 0.831i)9-s − 1.12i·11-s − 1.37i·13-s + (−0.728 + 1.36i)15-s − 0.904i·17-s − 1.85·19-s + (−0.0128 − 0.00686i)21-s − 0.392·23-s + 1.38·25-s + (0.995 − 0.0979i)27-s − 1.54·29-s − 0.458i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.290 + 0.956i$ Analytic conductor: $$22.6567$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{384} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :3/2),\ 0.290 + 0.956i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.396823894$$ $$L(\frac12)$$ $$\approx$$ $$1.396823894$$ $$L(\frac{5}{2})$$ 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 + (2.44 - 4.58i)T$$
good5 $$1 - 17.2T + 125T^{2}$$
7 $$1 - 0.269iT - 343T^{2}$$
11 $$1 + 41.1iT - 1.33e3T^{2}$$
13 $$1 + 64.6iT - 2.19e3T^{2}$$
17 $$1 + 63.4iT - 4.91e3T^{2}$$
19 $$1 + 154.T + 6.85e3T^{2}$$
23 $$1 + 43.3T + 1.21e4T^{2}$$
29 $$1 + 240.T + 2.43e4T^{2}$$
31 $$1 + 79.1iT - 2.97e4T^{2}$$
37 $$1 + 132. iT - 5.06e4T^{2}$$
41 $$1 - 101. iT - 6.89e4T^{2}$$
43 $$1 + 120.T + 7.95e4T^{2}$$
47 $$1 + 293.T + 1.03e5T^{2}$$
53 $$1 + 6.17T + 1.48e5T^{2}$$
59 $$1 - 45.8iT - 2.05e5T^{2}$$
61 $$1 + 651. iT - 2.26e5T^{2}$$
67 $$1 - 685.T + 3.00e5T^{2}$$
71 $$1 - 836.T + 3.57e5T^{2}$$
73 $$1 - 285.T + 3.89e5T^{2}$$
79 $$1 - 940. iT - 4.93e5T^{2}$$
83 $$1 + 1.01e3iT - 5.71e5T^{2}$$
89 $$1 - 432. iT - 7.04e5T^{2}$$
97 $$1 - 1.00e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$