# Properties

 Degree $2$ Conductor $1024$ Sign $-0.923 + 0.382i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.41 + 1.41i)3-s + (−1.41 − 1.41i)5-s + 4i·7-s − 1.00i·9-s + (1.41 + 1.41i)11-s + (−1.41 + 1.41i)13-s + 4.00·15-s + 2·17-s + (1.41 − 1.41i)19-s + (−5.65 − 5.65i)21-s + 4i·23-s − 0.999i·25-s + (−2.82 − 2.82i)27-s + (−4.24 + 4.24i)29-s − 4.00·33-s + ⋯
 L(s)  = 1 + (−0.816 + 0.816i)3-s + (−0.632 − 0.632i)5-s + 1.51i·7-s − 0.333i·9-s + (0.426 + 0.426i)11-s + (−0.392 + 0.392i)13-s + 1.03·15-s + 0.485·17-s + (0.324 − 0.324i)19-s + (−1.23 − 1.23i)21-s + 0.834i·23-s − 0.199i·25-s + (−0.544 − 0.544i)27-s + (−0.787 + 0.787i)29-s − 0.696·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1024$$    =    $$2^{10}$$ Sign: $-0.923 + 0.382i$ Motivic weight: $$1$$ Character: $\chi_{1024} (769, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1024,\ (\ :1/2),\ -0.923 + 0.382i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4023406400$$ $$L(\frac12)$$ $$\approx$$ $$0.4023406400$$ $$L(\frac{3}{2})$$ 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 + (1.41 - 1.41i)T - 3iT^{2}$$
5 $$1 + (1.41 + 1.41i)T + 5iT^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 + (-1.41 - 1.41i)T + 11iT^{2}$$
13 $$1 + (1.41 - 1.41i)T - 13iT^{2}$$
17 $$1 - 2T + 17T^{2}$$
19 $$1 + (-1.41 + 1.41i)T - 19iT^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 + (4.24 - 4.24i)T - 29iT^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + (7.07 + 7.07i)T + 37iT^{2}$$
41 $$1 - 6iT - 41T^{2}$$
43 $$1 + (4.24 + 4.24i)T + 43iT^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + (4.24 + 4.24i)T + 53iT^{2}$$
59 $$1 + (-9.89 - 9.89i)T + 59iT^{2}$$
61 $$1 + (-1.41 + 1.41i)T - 61iT^{2}$$
67 $$1 + (7.07 - 7.07i)T - 67iT^{2}$$
71 $$1 + 12iT - 71T^{2}$$
73 $$1 + 14iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 + (4.24 - 4.24i)T - 83iT^{2}$$
89 $$1 + 2iT - 89T^{2}$$
97 $$1 + 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$