# Properties

 Degree $2$ Conductor $384$ Sign $0.934 - 0.356i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.618 + 1.61i)3-s − 3.23i·5-s + 1.23i·7-s + (−2.23 + 2.00i)9-s + 5.23·11-s + 4.47·13-s + (5.23 − 2.00i)15-s + 2.47i·17-s − 0.763i·19-s + (−2.00 + 0.763i)21-s + 2.47·23-s − 5.47·25-s + (−4.61 − 2.38i)27-s − 4.76i·29-s + 5.23i·31-s + ⋯
 L(s)  = 1 + (0.356 + 0.934i)3-s − 1.44i·5-s + 0.467i·7-s + (−0.745 + 0.666i)9-s + 1.57·11-s + 1.24·13-s + (1.35 − 0.516i)15-s + 0.599i·17-s − 0.175i·19-s + (−0.436 + 0.166i)21-s + 0.515·23-s − 1.09·25-s + (−0.888 − 0.458i)27-s − 0.884i·29-s + 0.940i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.934 - 0.356i$ Motivic weight: $$1$$ Character: $\chi_{384} (383, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1/2),\ 0.934 - 0.356i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.60962 + 0.296948i$$ $$L(\frac12)$$ $$\approx$$ $$1.60962 + 0.296948i$$ $$L(\frac{3}{2})$$ 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 + (-0.618 - 1.61i)T$$
good5 $$1 + 3.23iT - 5T^{2}$$
7 $$1 - 1.23iT - 7T^{2}$$
11 $$1 - 5.23T + 11T^{2}$$
13 $$1 - 4.47T + 13T^{2}$$
17 $$1 - 2.47iT - 17T^{2}$$
19 $$1 + 0.763iT - 19T^{2}$$
23 $$1 - 2.47T + 23T^{2}$$
29 $$1 + 4.76iT - 29T^{2}$$
31 $$1 - 5.23iT - 31T^{2}$$
37 $$1 + 8.47T + 37T^{2}$$
41 $$1 + 6.47iT - 41T^{2}$$
43 $$1 - 7.23iT - 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + 3.23iT - 53T^{2}$$
59 $$1 + 1.23T + 59T^{2}$$
61 $$1 + 0.472T + 61T^{2}$$
67 $$1 + 9.70iT - 67T^{2}$$
71 $$1 + 15.4T + 71T^{2}$$
73 $$1 + 2T + 73T^{2}$$
79 $$1 - 0.291iT - 79T^{2}$$
83 $$1 - 2.76T + 83T^{2}$$
89 $$1 + 4iT - 89T^{2}$$
97 $$1 - 0.472T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$