# Properties

 Label 2-6-3.2-c10-0-3 Degree $2$ Conductor $6$ Sign $-0.566 + 0.824i$ Analytic cond. $3.81214$ Root an. cond. $1.95247$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 22.6i·2-s + (−200. − 137. i)3-s − 512.·4-s − 3.63e3i·5-s + (3.11e3 − 4.53e3i)6-s − 2.32e4·7-s − 1.15e4i·8-s + (2.11e4 + 5.51e4i)9-s + 8.21e4·10-s + 6.24e4i·11-s + (1.02e5 + 7.04e4i)12-s − 1.70e5·13-s − 5.25e5i·14-s + (−4.99e5 + 7.27e5i)15-s + 2.62e5·16-s − 2.66e6i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (−0.824 − 0.566i)3-s − 0.500·4-s − 1.16i·5-s + (0.400 − 0.582i)6-s − 1.38·7-s − 0.353i·8-s + (0.358 + 0.933i)9-s + 0.821·10-s + 0.387i·11-s + (0.412 + 0.283i)12-s − 0.458·13-s − 0.977i·14-s + (−0.657 + 0.957i)15-s + 0.250·16-s − 1.87i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6$$    =    $$2 \cdot 3$$ Sign: $-0.566 + 0.824i$ Analytic conductor: $$3.81214$$ Root analytic conductor: $$1.95247$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{6} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 6,\ (\ :5),\ -0.566 + 0.824i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.201661 - 0.383272i$$ $$L(\frac12)$$ $$\approx$$ $$0.201661 - 0.383272i$$ $$L(6)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 22.6iT$$
3 $$1 + (200. + 137. i)T$$
good5 $$1 + 3.63e3iT - 9.76e6T^{2}$$
7 $$1 + 2.32e4T + 2.82e8T^{2}$$
11 $$1 - 6.24e4iT - 2.59e10T^{2}$$
13 $$1 + 1.70e5T + 1.37e11T^{2}$$
17 $$1 + 2.66e6iT - 2.01e12T^{2}$$
19 $$1 - 7.66e5T + 6.13e12T^{2}$$
23 $$1 - 1.40e6iT - 4.14e13T^{2}$$
29 $$1 - 4.83e6iT - 4.20e14T^{2}$$
31 $$1 + 4.18e7T + 8.19e14T^{2}$$
37 $$1 - 5.01e7T + 4.80e15T^{2}$$
41 $$1 + 1.49e8iT - 1.34e16T^{2}$$
43 $$1 + 1.98e8T + 2.16e16T^{2}$$
47 $$1 + 1.55e8iT - 5.25e16T^{2}$$
53 $$1 - 4.21e7iT - 1.74e17T^{2}$$
59 $$1 - 2.92e8iT - 5.11e17T^{2}$$
61 $$1 + 5.30e8T + 7.13e17T^{2}$$
67 $$1 - 5.22e8T + 1.82e18T^{2}$$
71 $$1 - 5.71e8iT - 3.25e18T^{2}$$
73 $$1 - 2.18e9T + 4.29e18T^{2}$$
79 $$1 - 1.96e9T + 9.46e18T^{2}$$
83 $$1 + 2.18e9iT - 1.55e19T^{2}$$
89 $$1 + 2.38e8iT - 3.11e19T^{2}$$
97 $$1 + 8.84e9T + 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$