# Properties

 Degree $2$ Conductor $384$ Sign $0.985 - 0.169i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.41 − i)3-s + 2.82·5-s + 2.82i·7-s + (1.00 + 2.82i)9-s + 2i·11-s − 4i·13-s + (−4.00 − 2.82i)15-s + 5.65i·17-s + 2.82·19-s + (2.82 − 4.00i)21-s + 8·23-s + 3.00·25-s + (1.41 − 5.00i)27-s + 2.82·29-s − 8.48i·31-s + ⋯
 L(s)  = 1 + (−0.816 − 0.577i)3-s + 1.26·5-s + 1.06i·7-s + (0.333 + 0.942i)9-s + 0.603i·11-s − 1.10i·13-s + (−1.03 − 0.730i)15-s + 1.37i·17-s + 0.648·19-s + (0.617 − 0.872i)21-s + 1.66·23-s + 0.600·25-s + (0.272 − 0.962i)27-s + 0.525·29-s − 1.52i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $0.985 - 0.169i$ Motivic weight: $$1$$ Character: $\chi_{384} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :1/2),\ 0.985 - 0.169i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.30749 + 0.111352i$$ $$L(\frac12)$$ $$\approx$$ $$1.30749 + 0.111352i$$ $$L(\frac{3}{2})$$ 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 + (1.41 + i)T$$
good5 $$1 - 2.82T + 5T^{2}$$
7 $$1 - 2.82iT - 7T^{2}$$
11 $$1 - 2iT - 11T^{2}$$
13 $$1 + 4iT - 13T^{2}$$
17 $$1 - 5.65iT - 17T^{2}$$
19 $$1 - 2.82T + 19T^{2}$$
23 $$1 - 8T + 23T^{2}$$
29 $$1 - 2.82T + 29T^{2}$$
31 $$1 + 8.48iT - 31T^{2}$$
37 $$1 - 4iT - 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + 2.82T + 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 8.48T + 53T^{2}$$
59 $$1 - 6iT - 59T^{2}$$
61 $$1 + 4iT - 61T^{2}$$
67 $$1 + 14.1T + 67T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 - 2.82iT - 79T^{2}$$
83 $$1 + 6iT - 83T^{2}$$
89 $$1 + 5.65iT - 89T^{2}$$
97 $$1 + 6T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$