# Properties

 Degree $2$ Conductor $256$ Sign $0.707 + 0.707i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·3-s + 2i·5-s + 4·7-s − 9-s − 2i·11-s − 2i·13-s + 4·15-s − 2·17-s − 2i·19-s − 8i·21-s − 4·23-s + 25-s − 4i·27-s + 6i·29-s − 4·33-s + ⋯
 L(s)  = 1 − 1.15i·3-s + 0.894i·5-s + 1.51·7-s − 0.333·9-s − 0.603i·11-s − 0.554i·13-s + 1.03·15-s − 0.485·17-s − 0.458i·19-s − 1.74i·21-s − 0.834·23-s + 0.200·25-s − 0.769i·27-s + 1.11i·29-s − 0.696·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$256$$    =    $$2^{8}$$ Sign: $0.707 + 0.707i$ Motivic weight: $$1$$ Character: $\chi_{256} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 256,\ (\ :1/2),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.31847 - 0.546129i$$ $$L(\frac12)$$ $$\approx$$ $$1.31847 - 0.546129i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 2iT - 3T^{2}$$
5 $$1 - 2iT - 5T^{2}$$
7 $$1 - 4T + 7T^{2}$$
11 $$1 + 2iT - 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 + 2T + 17T^{2}$$
19 $$1 + 2iT - 19T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 - 6iT - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 10iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 - 6iT - 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 - 14iT - 59T^{2}$$
61 $$1 + 2iT - 61T^{2}$$
67 $$1 + 10iT - 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + 14T + 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 - 2T + 89T^{2}$$
97 $$1 + 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$