# Properties

 Label 2-2816-88.43-c1-0-79 Degree $2$ Conductor $2816$ Sign $0.707 + 0.707i$ Analytic cond. $22.4858$ Root an. cond. $4.74192$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.31·3-s − 3i·5-s + 8·9-s + 3.31·11-s − 9.94i·15-s + 3.31i·23-s − 4·25-s + 16.5·27-s + 9.94i·31-s + 11·33-s − 7i·37-s − 24i·45-s − 6.63i·47-s − 7·49-s − 6i·53-s + ⋯
 L(s)  = 1 + 1.91·3-s − 1.34i·5-s + 2.66·9-s + 1.00·11-s − 2.56i·15-s + 0.691i·23-s − 0.800·25-s + 3.19·27-s + 1.78i·31-s + 1.91·33-s − 1.15i·37-s − 3.57i·45-s − 0.967i·47-s − 49-s − 0.824i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2816$$    =    $$2^{8} \cdot 11$$ Sign: $0.707 + 0.707i$ Analytic conductor: $$22.4858$$ Root analytic conductor: $$4.74192$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2816} (1407, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2816,\ (\ :1/2),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.164253662$$ $$L(\frac12)$$ $$\approx$$ $$4.164253662$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
11 $$1 - 3.31T$$
good3 $$1 - 3.31T + 3T^{2}$$
5 $$1 + 3iT - 5T^{2}$$
7 $$1 + 7T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 - 3.31iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 9.94iT - 31T^{2}$$
37 $$1 + 7iT - 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 6.63iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 3.31T + 59T^{2}$$
61 $$1 + 61T^{2}$$
67 $$1 + 9.94T + 67T^{2}$$
71 $$1 - 16.5iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 - 9T + 89T^{2}$$
97 $$1 + 17T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$