# Properties

 Degree $2$ Conductor $256$ Sign $0.555 - 0.831i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 + 1.70i)3-s + (3.12 + 1.29i)5-s + (−1 − i)7-s + (−0.292 + 0.292i)9-s + (0.121 − 0.292i)11-s + (−1.70 + 0.707i)13-s + 6.24i·15-s − 2.82i·17-s + (−5.53 + 2.29i)19-s + (0.999 − 2.41i)21-s + (−0.171 + 0.171i)23-s + (4.53 + 4.53i)25-s + (4.41 + 1.82i)27-s + (−1.12 − 2.70i)29-s + 4·31-s + ⋯
 L(s)  = 1 + (0.408 + 0.985i)3-s + (1.39 + 0.578i)5-s + (−0.377 − 0.377i)7-s + (−0.0976 + 0.0976i)9-s + (0.0365 − 0.0883i)11-s + (−0.473 + 0.196i)13-s + 1.61i·15-s − 0.685i·17-s + (−1.26 + 0.526i)19-s + (0.218 − 0.526i)21-s + (−0.0357 + 0.0357i)23-s + (0.907 + 0.907i)25-s + (0.849 + 0.351i)27-s + (−0.208 − 0.502i)29-s + 0.718·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$256$$    =    $$2^{8}$$ Sign: $0.555 - 0.831i$ Motivic weight: $$1$$ Character: $\chi_{256} (225, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 256,\ (\ :1/2),\ 0.555 - 0.831i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.45245 + 0.776355i$$ $$L(\frac12)$$ $$\approx$$ $$1.45245 + 0.776355i$$ $$L(\frac{3}{2})$$ 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 + (-0.707 - 1.70i)T + (-2.12 + 2.12i)T^{2}$$
5 $$1 + (-3.12 - 1.29i)T + (3.53 + 3.53i)T^{2}$$
7 $$1 + (1 + i)T + 7iT^{2}$$
11 $$1 + (-0.121 + 0.292i)T + (-7.77 - 7.77i)T^{2}$$
13 $$1 + (1.70 - 0.707i)T + (9.19 - 9.19i)T^{2}$$
17 $$1 + 2.82iT - 17T^{2}$$
19 $$1 + (5.53 - 2.29i)T + (13.4 - 13.4i)T^{2}$$
23 $$1 + (0.171 - 0.171i)T - 23iT^{2}$$
29 $$1 + (1.12 + 2.70i)T + (-20.5 + 20.5i)T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 + (1.70 + 0.707i)T + (26.1 + 26.1i)T^{2}$$
41 $$1 + (5.82 - 5.82i)T - 41iT^{2}$$
43 $$1 + (-3.29 + 7.94i)T + (-30.4 - 30.4i)T^{2}$$
47 $$1 + 11.6iT - 47T^{2}$$
53 $$1 + (3.12 - 7.53i)T + (-37.4 - 37.4i)T^{2}$$
59 $$1 + (6.12 + 2.53i)T + (41.7 + 41.7i)T^{2}$$
61 $$1 + (0.292 + 0.707i)T + (-43.1 + 43.1i)T^{2}$$
67 $$1 + (-1.53 - 3.70i)T + (-47.3 + 47.3i)T^{2}$$
71 $$1 + (-0.171 - 0.171i)T + 71iT^{2}$$
73 $$1 + (-7 + 7i)T - 73iT^{2}$$
79 $$1 - 6iT - 79T^{2}$$
83 $$1 + (-6.12 + 2.53i)T + (58.6 - 58.6i)T^{2}$$
89 $$1 + (2.65 + 2.65i)T + 89iT^{2}$$
97 $$1 + 1.51T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$