# Properties

 Label 2-1400-56.13-c0-0-1 Degree $2$ Conductor $1400$ Sign $1$ 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 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$1400$$    =    $$2^{3} \cdot 5^{2} \cdot 7$$ Sign: $1$ Analytic conductor: $$0.698691$$ Root analytic conductor: $$0.835877$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1400} (1301, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1400,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8170917516$$ $$L(\frac12)$$ $$\approx$$ $$0.8170917516$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 + 0.866i)T$$
5 $$1$$
7 $$1 - T$$
good3 $$1 + T^{2}$$
11 $$1 - 1.73iT - T^{2}$$
13 $$1 + T^{2}$$
17 $$1 - T^{2}$$
19 $$1 + T^{2}$$
23 $$1 - T + T^{2}$$
29 $$1 - 1.73iT - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - 1.73iT - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.73iT - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + 1.73iT - T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T + T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$