# Properties

 Label 2-2e8-8.5-c5-0-6 Degree $2$ Conductor $256$ Sign $-0.707 - 0.707i$ Analytic cond. $41.0582$ Root an. cond. $6.40767$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 20i·3-s − 74i·5-s − 24·7-s − 157·9-s − 124i·11-s − 478i·13-s + 1.48e3·15-s − 1.19e3·17-s + 3.04e3i·19-s − 480i·21-s + 184·23-s − 2.35e3·25-s + 1.72e3i·27-s + 3.28e3i·29-s + 5.72e3·31-s + ⋯
 L(s)  = 1 + 1.28i·3-s − 1.32i·5-s − 0.185·7-s − 0.646·9-s − 0.308i·11-s − 0.784i·13-s + 1.69·15-s − 1.00·17-s + 1.93i·19-s − 0.237i·21-s + 0.0725·23-s − 0.752·25-s + 0.454i·27-s + 0.724i·29-s + 1.07·31-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: $-0.707 - 0.707i$ Analytic conductor: $$41.0582$$ Root analytic conductor: $$6.40767$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes 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

 $$L(3)$$ $$\approx$$ $$1.092727895$$ $$L(\frac12)$$ $$\approx$$ $$1.092727895$$ $$L(\frac{7}{2})$$ 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 - 20iT - 243T^{2}$$
5 $$1 + 74iT - 3.12e3T^{2}$$
7 $$1 + 24T + 1.68e4T^{2}$$
11 $$1 + 124iT - 1.61e5T^{2}$$
13 $$1 + 478iT - 3.71e5T^{2}$$
17 $$1 + 1.19e3T + 1.41e6T^{2}$$
19 $$1 - 3.04e3iT - 2.47e6T^{2}$$
23 $$1 - 184T + 6.43e6T^{2}$$
29 $$1 - 3.28e3iT - 2.05e7T^{2}$$
31 $$1 - 5.72e3T + 2.86e7T^{2}$$
37 $$1 - 1.03e4iT - 6.93e7T^{2}$$
41 $$1 - 8.88e3T + 1.15e8T^{2}$$
43 $$1 - 9.18e3iT - 1.47e8T^{2}$$
47 $$1 + 2.36e4T + 2.29e8T^{2}$$
53 $$1 - 1.16e4iT - 4.18e8T^{2}$$
59 $$1 + 1.68e4iT - 7.14e8T^{2}$$
61 $$1 - 1.84e4iT - 8.44e8T^{2}$$
67 $$1 + 1.55e4iT - 1.35e9T^{2}$$
71 $$1 + 3.19e4T + 1.80e9T^{2}$$
73 $$1 - 4.88e3T + 2.07e9T^{2}$$
79 $$1 + 4.45e4T + 3.07e9T^{2}$$
83 $$1 - 6.73e4iT - 3.93e9T^{2}$$
89 $$1 + 7.19e4T + 5.58e9T^{2}$$
97 $$1 - 4.88e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$