# Properties

 Label 2-896-8.5-c1-0-21 Degree $2$ Conductor $896$ Sign $-1$ Analytic cond. $7.15459$ Root an. cond. $2.67480$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.61i·3-s + 2.61i·5-s − 7-s − 3.82·9-s − 2.16i·11-s − 0.448i·13-s + 6.82·15-s − 7.65·17-s − 4.77i·19-s + 2.61i·21-s − 6.82·23-s − 1.82·25-s + 2.16i·27-s − 9.55i·29-s − 5.65·31-s + ⋯
 L(s)  = 1 − 1.50i·3-s + 1.16i·5-s − 0.377·7-s − 1.27·9-s − 0.652i·11-s − 0.124i·13-s + 1.76·15-s − 1.85·17-s − 1.09i·19-s + 0.570i·21-s − 1.42·23-s − 0.365·25-s + 0.416i·27-s − 1.77i·29-s − 1.01·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$896$$    =    $$2^{7} \cdot 7$$ Sign: $-1$ Analytic conductor: $$7.15459$$ Root analytic conductor: $$2.67480$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{896} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 896,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.612968i$$ $$L(\frac12)$$ $$\approx$$ $$0.612968i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + T$$
good3 $$1 + 2.61iT - 3T^{2}$$
5 $$1 - 2.61iT - 5T^{2}$$
11 $$1 + 2.16iT - 11T^{2}$$
13 $$1 + 0.448iT - 13T^{2}$$
17 $$1 + 7.65T + 17T^{2}$$
19 $$1 + 4.77iT - 19T^{2}$$
23 $$1 + 6.82T + 23T^{2}$$
29 $$1 + 9.55iT - 29T^{2}$$
31 $$1 + 5.65T + 31T^{2}$$
37 $$1 - 5.22iT - 37T^{2}$$
41 $$1 + 3.65T + 41T^{2}$$
43 $$1 - 2.16iT - 43T^{2}$$
47 $$1 - 8T + 47T^{2}$$
53 $$1 + 10.4iT - 53T^{2}$$
59 $$1 - 0.448iT - 59T^{2}$$
61 $$1 - 12.1iT - 61T^{2}$$
67 $$1 + 3.06iT - 67T^{2}$$
71 $$1 + 2.34T + 71T^{2}$$
73 $$1 - 11.6T + 73T^{2}$$
79 $$1 - 2.34T + 79T^{2}$$
83 $$1 - 13.0iT - 83T^{2}$$
89 $$1 + 2T + 89T^{2}$$
97 $$1 + 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$