# Properties

 Label 2-2e9-16.13-c1-0-9 Degree $2$ Conductor $512$ Sign $0.382 + 0.923i$ Analytic cond. $4.08834$ Root an. cond. $2.02196$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − i)5-s − 3i·9-s + (−5 − 5i)13-s + 8·17-s + 3i·25-s + (−3 − 3i)29-s + (7 − 7i)37-s − 8i·41-s + (−3 − 3i)45-s + 7·49-s + (−9 + 9i)53-s + (11 + 11i)61-s − 10·65-s + 6i·73-s − 9·81-s + ⋯
 L(s)  = 1 + (0.447 − 0.447i)5-s − i·9-s + (−1.38 − 1.38i)13-s + 1.94·17-s + 0.600i·25-s + (−0.557 − 0.557i)29-s + (1.15 − 1.15i)37-s − 1.24i·41-s + (−0.447 − 0.447i)45-s + 49-s + (−1.23 + 1.23i)53-s + (1.40 + 1.40i)61-s − 1.24·65-s + 0.702i·73-s − 81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$512$$    =    $$2^{9}$$ Sign: $0.382 + 0.923i$ Analytic conductor: $$4.08834$$ Root analytic conductor: $$2.02196$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{512} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 512,\ (\ :1/2),\ 0.382 + 0.923i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.18874 - 0.794291i$$ $$L(\frac12)$$ $$\approx$$ $$1.18874 - 0.794291i$$ $$L(\frac{3}{2})$$ 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 + 3iT^{2}$$
5 $$1 + (-1 + i)T - 5iT^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 - 11iT^{2}$$
13 $$1 + (5 + 5i)T + 13iT^{2}$$
17 $$1 - 8T + 17T^{2}$$
19 $$1 + 19iT^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + (3 + 3i)T + 29iT^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + (-7 + 7i)T - 37iT^{2}$$
41 $$1 + 8iT - 41T^{2}$$
43 $$1 - 43iT^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + (9 - 9i)T - 53iT^{2}$$
59 $$1 - 59iT^{2}$$
61 $$1 + (-11 - 11i)T + 61iT^{2}$$
67 $$1 + 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 - 6iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 - 10iT - 89T^{2}$$
97 $$1 + 8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$