# Properties

 Label 2-700-7.6-c0-0-0 Degree $2$ Conductor $700$ Sign $-i$ Analytic cond. $0.349345$ Root an. cond. $0.591054$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + i·7-s − 11-s + i·13-s − i·17-s − 21-s + i·27-s + 29-s − i·33-s − 39-s − i·47-s − 49-s + 51-s + 2·71-s − 2i·73-s + ⋯
 L(s)  = 1 + i·3-s + i·7-s − 11-s + i·13-s − i·17-s − 21-s + i·27-s + 29-s − i·33-s − 39-s − i·47-s − 49-s + 51-s + 2·71-s − 2i·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$700$$    =    $$2^{2} \cdot 5^{2} \cdot 7$$ Sign: $-i$ Analytic conductor: $$0.349345$$ Root analytic conductor: $$0.591054$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{700} (601, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 700,\ (\ :0),\ -i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9140492637$$ $$L(\frac12)$$ $$\approx$$ $$0.9140492637$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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