# Properties

 Label 2-117-13.12-c3-0-15 Degree $2$ Conductor $117$ Sign $-0.719 - 0.694i$ 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 − 4.54i·2-s − 12.6·4-s − 12.9i·5-s − 16.7i·7-s + 21.2i·8-s − 58.7·10-s + 24.9i·11-s + (33.7 + 32.5i)13-s − 76.0·14-s − 4.67·16-s − 134.·17-s − 14.9i·19-s + 163. i·20-s + 113.·22-s + 72·23-s + ⋯
 L(s)  = 1 − 1.60i·2-s − 1.58·4-s − 1.15i·5-s − 0.903i·7-s + 0.940i·8-s − 1.85·10-s + 0.683i·11-s + (0.719 + 0.694i)13-s − 1.45·14-s − 0.0731·16-s − 1.91·17-s − 0.180i·19-s + 1.83i·20-s + 1.09·22-s + 0.652·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$117$$    =    $$3^{2} \cdot 13$$ Sign: $-0.719 - 0.694i$ 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: $$(2,\ 117,\ (\ :3/2),\ -0.719 - 0.694i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.444339 + 1.09968i$$ $$L(\frac12)$$ $$\approx$$ $$0.444339 + 1.09968i$$ $$L(\frac{5}{2})$$ 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 + (-33.7 - 32.5i)T$$
good2 $$1 + 4.54iT - 8T^{2}$$
5 $$1 + 12.9iT - 125T^{2}$$
7 $$1 + 16.7iT - 343T^{2}$$
11 $$1 - 24.9iT - 1.33e3T^{2}$$
17 $$1 + 134.T + 4.91e3T^{2}$$
19 $$1 + 14.9iT - 6.85e3T^{2}$$
23 $$1 - 72T + 1.21e4T^{2}$$
29 $$1 - 206.T + 2.43e4T^{2}$$
31 $$1 + 249. iT - 2.97e4T^{2}$$
37 $$1 + 293. iT - 5.06e4T^{2}$$
41 $$1 + 250. iT - 6.89e4T^{2}$$
43 $$1 + 432.T + 7.95e4T^{2}$$
47 $$1 + 159. iT - 1.03e5T^{2}$$
53 $$1 - 194.T + 1.48e5T^{2}$$
59 $$1 - 232. iT - 2.05e5T^{2}$$
61 $$1 + 185.T + 2.26e5T^{2}$$
67 $$1 + 39.4iT - 3.00e5T^{2}$$
71 $$1 + 920. iT - 3.57e5T^{2}$$
73 $$1 - 549. iT - 3.89e5T^{2}$$
79 $$1 - 933.T + 4.93e5T^{2}$$
83 $$1 - 1.09e3iT - 5.71e5T^{2}$$
89 $$1 + 532. iT - 7.04e5T^{2}$$
97 $$1 + 362. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$