# Properties

 Label 2-700-5.4-c5-0-19 Degree $2$ Conductor $700$ Sign $-0.894 + 0.447i$ Analytic cond. $112.268$ Root an. cond. $10.5956$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 26i·3-s + 49i·7-s − 433·9-s + 8·11-s + 684i·13-s + 2.21e3i·17-s + 2.69e3·19-s − 1.27e3·21-s + 3.34e3i·23-s − 4.94e3i·27-s + 3.25e3·29-s + 4.78e3·31-s + 208i·33-s + 1.14e4i·37-s − 1.77e4·39-s + ⋯
 L(s)  = 1 + 1.66i·3-s + 0.377i·7-s − 1.78·9-s + 0.0199·11-s + 1.12i·13-s + 1.86i·17-s + 1.71·19-s − 0.630·21-s + 1.31i·23-s − 1.30i·27-s + 0.718·29-s + 0.894·31-s + 0.0332i·33-s + 1.37i·37-s − 1.87·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$700$$    =    $$2^{2} \cdot 5^{2} \cdot 7$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$112.268$$ Root analytic conductor: $$10.5956$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{700} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 700,\ (\ :5/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.187914188$$ $$L(\frac12)$$ $$\approx$$ $$2.187914188$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1 - 49iT$$
good3 $$1 - 26iT - 243T^{2}$$
11 $$1 - 8T + 1.61e5T^{2}$$
13 $$1 - 684iT - 3.71e5T^{2}$$
17 $$1 - 2.21e3iT - 1.41e6T^{2}$$
19 $$1 - 2.69e3T + 2.47e6T^{2}$$
23 $$1 - 3.34e3iT - 6.43e6T^{2}$$
29 $$1 - 3.25e3T + 2.05e7T^{2}$$
31 $$1 - 4.78e3T + 2.86e7T^{2}$$
37 $$1 - 1.14e4iT - 6.93e7T^{2}$$
41 $$1 - 1.33e4T + 1.15e8T^{2}$$
43 $$1 + 928iT - 1.47e8T^{2}$$
47 $$1 + 1.21e3iT - 2.29e8T^{2}$$
53 $$1 - 1.31e4iT - 4.18e8T^{2}$$
59 $$1 + 3.47e4T + 7.14e8T^{2}$$
61 $$1 + 1.03e3T + 8.44e8T^{2}$$
67 $$1 + 1.01e4iT - 1.35e9T^{2}$$
71 $$1 - 6.27e4T + 1.80e9T^{2}$$
73 $$1 + 1.89e4iT - 2.07e9T^{2}$$
79 $$1 + 1.14e4T + 3.07e9T^{2}$$
83 $$1 - 8.89e4iT - 3.93e9T^{2}$$
89 $$1 + 1.97e4T + 5.58e9T^{2}$$
97 $$1 + 1.70e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$