# Properties

 Label 2-1170-13.10-c1-0-8 Degree $2$ Conductor $1170$ Sign $0.711 + 0.702i$ Analytic cond. $9.34249$ Root an. cond. $3.05654$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + (−1.5 − 0.866i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (−2.59 + 1.5i)11-s + (3.5 + 0.866i)13-s + 1.73·14-s + (−0.5 − 0.866i)16-s + (4.5 + 2.59i)19-s + (−0.866 − 0.499i)20-s + (1.5 − 2.59i)22-s + (−1.73 − 3i)23-s − 25-s + (−3.46 + i)26-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.566 − 0.327i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (−0.783 + 0.452i)11-s + (0.970 + 0.240i)13-s + 0.462·14-s + (−0.125 − 0.216i)16-s + (1.03 + 0.596i)19-s + (−0.193 − 0.111i)20-s + (0.319 − 0.553i)22-s + (−0.361 − 0.625i)23-s − 0.200·25-s + (−0.679 + 0.196i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1170$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 13$$ Sign: $0.711 + 0.702i$ Analytic conductor: $$9.34249$$ Root analytic conductor: $$3.05654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1170} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1170,\ (\ :1/2),\ 0.711 + 0.702i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9984604276$$ $$L(\frac12)$$ $$\approx$$ $$0.9984604276$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1$$
5 $$1 + iT$$
13 $$1 + (-3.5 - 0.866i)T$$
good7 $$1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3.46 - 6i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 10.3iT - 31T^{2}$$
37 $$1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 + (-5.19 + 3i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 9iT - 47T^{2}$$
53 $$1 + 12.1T + 53T^{2}$$
59 $$1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + 6.92iT - 73T^{2}$$
79 $$1 - 10T + 79T^{2}$$
83 $$1 + 12iT - 83T^{2}$$
89 $$1 + (-12.9 + 7.5i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$