Properties

 Degree $2$ Conductor $384$ Sign $i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 1.73i·3-s − 2.82·5-s − 4.89i·7-s − 2.99·9-s − 3.46i·11-s − 4.89i·15-s + 8.48·21-s + 3.00·25-s − 5.19i·27-s − 2.82·29-s − 4.89i·31-s + 5.99·33-s + 13.8i·35-s + 8.48·45-s − 16.9·49-s + ⋯
 L(s)  = 1 + 0.999i·3-s − 1.26·5-s − 1.85i·7-s − 0.999·9-s − 1.04i·11-s − 1.26i·15-s + 1.85·21-s + 0.600·25-s − 0.999i·27-s − 0.525·29-s − 0.879i·31-s + 1.04·33-s + 2.34i·35-s + 1.26·45-s − 2.42·49-s + ⋯

Functional equation

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

Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $i$ Motivic weight: $$1$$ Character: $\chi_{384} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1/2),\ i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.459238 - 0.459238i$$ $$L(\frac12)$$ $$\approx$$ $$0.459238 - 0.459238i$$ $$L(\frac{3}{2})$$ 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 - 1.73iT$$
good5 $$1 + 2.82T + 5T^{2}$$
7 $$1 + 4.89iT - 7T^{2}$$
11 $$1 + 3.46iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + 2.82T + 29T^{2}$$
31 $$1 + 4.89iT - 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 14.1T + 53T^{2}$$
59 $$1 + 10.3iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 14T + 73T^{2}$$
79 $$1 - 14.6iT - 79T^{2}$$
83 $$1 + 17.3iT - 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$