# Properties

 Degree $2$ Conductor $1024$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.80·3-s − 0.844·5-s − 29.0·7-s − 19.1·9-s − 17.1·11-s − 68.6·13-s + 2.36·15-s − 86.7·17-s − 77.5·19-s + 81.5·21-s + 70.2·23-s − 124.·25-s + 129.·27-s − 89.6·29-s − 8.86·31-s + 48.1·33-s + 24.5·35-s − 30.7·37-s + 192.·39-s − 153.·41-s + 171.·43-s + 16.1·45-s + 99.9·47-s + 502.·49-s + 243.·51-s − 550.·53-s + 14.4·55-s + ⋯
 L(s)  = 1 − 0.539·3-s − 0.0754·5-s − 1.57·7-s − 0.708·9-s − 0.470·11-s − 1.46·13-s + 0.0407·15-s − 1.23·17-s − 0.936·19-s + 0.847·21-s + 0.636·23-s − 0.994·25-s + 0.922·27-s − 0.574·29-s − 0.0513·31-s + 0.253·33-s + 0.118·35-s − 0.136·37-s + 0.791·39-s − 0.583·41-s + 0.606·43-s + 0.0534·45-s + 0.310·47-s + 1.46·49-s + 0.667·51-s − 1.42·53-s + 0.0354·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1024$$    =    $$2^{10}$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{1024} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1024,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.002746848088$$ $$L(\frac12)$$ $$\approx$$ $$0.002746848088$$ $$L(\frac{5}{2})$$ 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 + 2.80T + 27T^{2}$$
5 $$1 + 0.844T + 125T^{2}$$
7 $$1 + 29.0T + 343T^{2}$$
11 $$1 + 17.1T + 1.33e3T^{2}$$
13 $$1 + 68.6T + 2.19e3T^{2}$$
17 $$1 + 86.7T + 4.91e3T^{2}$$
19 $$1 + 77.5T + 6.85e3T^{2}$$
23 $$1 - 70.2T + 1.21e4T^{2}$$
29 $$1 + 89.6T + 2.43e4T^{2}$$
31 $$1 + 8.86T + 2.97e4T^{2}$$
37 $$1 + 30.7T + 5.06e4T^{2}$$
41 $$1 + 153.T + 6.89e4T^{2}$$
43 $$1 - 171.T + 7.95e4T^{2}$$
47 $$1 - 99.9T + 1.03e5T^{2}$$
53 $$1 + 550.T + 1.48e5T^{2}$$
59 $$1 + 459.T + 2.05e5T^{2}$$
61 $$1 - 0.479T + 2.26e5T^{2}$$
67 $$1 + 799.T + 3.00e5T^{2}$$
71 $$1 - 419.T + 3.57e5T^{2}$$
73 $$1 - 374.T + 3.89e5T^{2}$$
79 $$1 - 705.T + 4.93e5T^{2}$$
83 $$1 + 1.33e3T + 5.71e5T^{2}$$
89 $$1 - 4.72T + 7.04e5T^{2}$$
97 $$1 - 379.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$