# Properties

 Label 2-1400-280.83-c0-0-4 Degree $2$ Conductor $1400$ Sign $0.793 + 0.608i$ Analytic cond. $0.698691$ Root an. cond. $0.835877$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + 11-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.258 − 0.965i)22-s + (−0.707 + 0.707i)23-s + (−0.258 − 0.965i)28-s + 1.73·29-s + (0.965 − 0.258i)32-s + (0.499 − 0.866i)36-s + (−0.707 − 0.707i)37-s + ⋯
 L(s)  = 1 + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + 11-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.258 − 0.965i)22-s + (−0.707 + 0.707i)23-s + (−0.258 − 0.965i)28-s + 1.73·29-s + (0.965 − 0.258i)32-s + (0.499 − 0.866i)36-s + (−0.707 − 0.707i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1400$$    =    $$2^{3} \cdot 5^{2} \cdot 7$$ Sign: $0.793 + 0.608i$ Analytic conductor: $$0.698691$$ Root analytic conductor: $$0.835877$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1400} (643, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1400,\ (\ :0),\ 0.793 + 0.608i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.241942799$$ $$L(\frac12)$$ $$\approx$$ $$1.241942799$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.258 + 0.965i)T$$
5 $$1$$
7 $$1 + (-0.707 - 0.707i)T$$
good3 $$1 - iT^{2}$$
11 $$1 - T + T^{2}$$
13 $$1 + iT^{2}$$
17 $$1 + iT^{2}$$
19 $$1 + T^{2}$$
23 $$1 + (0.707 - 0.707i)T - iT^{2}$$
29 $$1 - 1.73T + T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (0.707 + 0.707i)T + iT^{2}$$
41 $$1 - T^{2}$$
43 $$1 + (1.22 + 1.22i)T + iT^{2}$$
47 $$1 - iT^{2}$$
53 $$1 + (-1.41 + 1.41i)T - iT^{2}$$
59 $$1 + T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (1.22 - 1.22i)T - iT^{2}$$
71 $$1 + 1.73iT - T^{2}$$
73 $$1 - iT^{2}$$
79 $$1 + 1.73T + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$