# Properties

 Label 2-117-13.12-c3-0-9 Degree $2$ Conductor $117$ Sign $1$ 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 + 4.42i·2-s − 11.6·4-s − 20.3i·5-s − 15.9i·8-s + 90.2·10-s − 70.0i·11-s + 46.8·13-s − 22.1·16-s + 236. i·20-s + 310.·22-s − 290.·25-s + 207. i·26-s − 225. i·32-s − 325.·40-s − 486. i·41-s + ⋯
 L(s)  = 1 + 1.56i·2-s − 1.45·4-s − 1.82i·5-s − 0.705i·8-s + 2.85·10-s − 1.91i·11-s + 1.00·13-s − 0.346·16-s + 2.64i·20-s + 3.00·22-s − 2.32·25-s + 1.56i·26-s − 1.24i·32-s − 1.28·40-s − 1.85i·41-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$117$$    =    $$3^{2} \cdot 13$$ Sign: $1$ 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: $$(2,\ 117,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.33582$$ $$L(\frac12)$$ $$\approx$$ $$1.33582$$ $$L(\frac{5}{2})$$ 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 - 46.8T$$
good2 $$1 - 4.42iT - 8T^{2}$$
5 $$1 + 20.3iT - 125T^{2}$$
7 $$1 - 343T^{2}$$
11 $$1 + 70.0iT - 1.33e3T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 + 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 - 5.06e4T^{2}$$
41 $$1 + 486. iT - 6.89e4T^{2}$$
43 $$1 - 452T + 7.95e4T^{2}$$
47 $$1 - 71.1iT - 1.03e5T^{2}$$
53 $$1 + 1.48e5T^{2}$$
59 $$1 - 696. iT - 2.05e5T^{2}$$
61 $$1 + 944.T + 2.26e5T^{2}$$
67 $$1 - 3.00e5T^{2}$$
71 $$1 + 123. iT - 3.57e5T^{2}$$
73 $$1 - 3.89e5T^{2}$$
79 $$1 - 418.T + 4.93e5T^{2}$$
83 $$1 - 1.50e3iT - 5.71e5T^{2}$$
89 $$1 + 155. iT - 7.04e5T^{2}$$
97 $$1 - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$