# Properties

 Degree $2$ Conductor $256$ Sign $0.707 - 0.707i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 8i·3-s − 14i·5-s − 208·7-s + 179·9-s − 536i·11-s + 694i·13-s + 112·15-s − 1.27e3·17-s − 1.11e3i·19-s − 1.66e3i·21-s + 3.21e3·23-s + 2.92e3·25-s + 3.37e3i·27-s + 2.91e3i·29-s + 2.62e3·31-s + ⋯
 L(s)  = 1 + 0.513i·3-s − 0.250i·5-s − 1.60·7-s + 0.736·9-s − 1.33i·11-s + 1.13i·13-s + 0.128·15-s − 1.07·17-s − 0.706i·19-s − 0.823i·21-s + 1.26·23-s + 0.937·25-s + 0.891i·27-s + 0.644i·29-s + 0.490·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, 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: $$5$$ Character: $\chi_{256} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 256,\ (\ :5/2),\ 0.707 - 0.707i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.532736939$$ $$L(\frac12)$$ $$\approx$$ $$1.532736939$$ $$L(\frac{7}{2})$$ 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 - 8iT - 243T^{2}$$
5 $$1 + 14iT - 3.12e3T^{2}$$
7 $$1 + 208T + 1.68e4T^{2}$$
11 $$1 + 536iT - 1.61e5T^{2}$$
13 $$1 - 694iT - 3.71e5T^{2}$$
17 $$1 + 1.27e3T + 1.41e6T^{2}$$
19 $$1 + 1.11e3iT - 2.47e6T^{2}$$
23 $$1 - 3.21e3T + 6.43e6T^{2}$$
29 $$1 - 2.91e3iT - 2.05e7T^{2}$$
31 $$1 - 2.62e3T + 2.86e7T^{2}$$
37 $$1 - 9.45e3iT - 6.93e7T^{2}$$
41 $$1 + 170T + 1.15e8T^{2}$$
43 $$1 + 1.99e4iT - 1.47e8T^{2}$$
47 $$1 + 32T + 2.29e8T^{2}$$
53 $$1 - 2.21e4iT - 4.18e8T^{2}$$
59 $$1 - 4.14e4iT - 7.14e8T^{2}$$
61 $$1 - 1.54e4iT - 8.44e8T^{2}$$
67 $$1 - 2.07e4iT - 1.35e9T^{2}$$
71 $$1 - 2.85e4T + 1.80e9T^{2}$$
73 $$1 - 5.36e4T + 2.07e9T^{2}$$
79 $$1 - 6.91e4T + 3.07e9T^{2}$$
83 $$1 - 3.78e4iT - 3.93e9T^{2}$$
89 $$1 - 1.26e5T + 5.58e9T^{2}$$
97 $$1 - 6.22e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$