# Properties

 Label 2-117-13.12-c3-0-13 Degree $2$ Conductor $117$ Sign $-0.159 + 0.987i$ Analytic cond. $6.90322$ Root an. cond. $2.62739$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.32i·2-s + 6.23·4-s − 15.4i·5-s + 7.96i·7-s − 18.9i·8-s − 20.4·10-s + 12.7i·11-s + (7.47 − 46.2i)13-s + 10.5·14-s + 24.8·16-s − 54·17-s − 84.5i·19-s − 96.2i·20-s + 16.9·22-s + 122.·23-s + ⋯
 L(s)  = 1 − 0.469i·2-s + 0.779·4-s − 1.37i·5-s + 0.430i·7-s − 0.835i·8-s − 0.647·10-s + 0.350i·11-s + (0.159 − 0.987i)13-s + 0.201·14-s + 0.387·16-s − 0.770·17-s − 1.02i·19-s − 1.07i·20-s + 0.164·22-s + 1.11·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$117$$    =    $$3^{2} \cdot 13$$ Sign: $-0.159 + 0.987i$ Analytic conductor: $$6.90322$$ Root analytic conductor: $$2.62739$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{117} (64, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 117,\ (\ :3/2),\ -0.159 + 0.987i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.20047 - 1.40998i$$ $$L(\frac12)$$ $$\approx$$ $$1.20047 - 1.40998i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
13 $$1 + (-7.47 + 46.2i)T$$
good2 $$1 + 1.32iT - 8T^{2}$$
5 $$1 + 15.4iT - 125T^{2}$$
7 $$1 - 7.96iT - 343T^{2}$$
11 $$1 - 12.7iT - 1.33e3T^{2}$$
17 $$1 + 54T + 4.91e3T^{2}$$
19 $$1 + 84.5iT - 6.85e3T^{2}$$
23 $$1 - 122.T + 1.21e4T^{2}$$
29 $$1 + 140.T + 2.43e4T^{2}$$
31 $$1 + 116. iT - 2.97e4T^{2}$$
37 $$1 - 433. iT - 5.06e4T^{2}$$
41 $$1 - 205. iT - 6.89e4T^{2}$$
43 $$1 - 418.T + 7.95e4T^{2}$$
47 $$1 - 485. iT - 1.03e5T^{2}$$
53 $$1 - 674.T + 1.48e5T^{2}$$
59 $$1 - 186. iT - 2.05e5T^{2}$$
61 $$1 + 671.T + 2.26e5T^{2}$$
67 $$1 - 14.0iT - 3.00e5T^{2}$$
71 $$1 - 346. iT - 3.57e5T^{2}$$
73 $$1 - 832. iT - 3.89e5T^{2}$$
79 $$1 + 335.T + 4.93e5T^{2}$$
83 $$1 + 568. iT - 5.71e5T^{2}$$
89 $$1 + 236. iT - 7.04e5T^{2}$$
97 $$1 + 1.27e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$