# Properties

 Label 2-39-13.12-c3-0-3 Degree $2$ Conductor $39$ Sign $0.847 - 0.531i$ Analytic cond. $2.30107$ Root an. cond. $1.51692$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.52i·2-s + 3·3-s + 5.67·4-s − 9.65i·5-s + 4.57i·6-s + 22.3i·7-s + 20.8i·8-s + 9·9-s + 14.7·10-s − 50.3i·11-s + 17.0·12-s + (−39.7 + 24.8i)13-s − 34.0·14-s − 28.9i·15-s + 13.6·16-s − 86.1·17-s + ⋯
 L(s)  = 1 + 0.538i·2-s + 0.577·3-s + 0.709·4-s − 0.863i·5-s + 0.310i·6-s + 1.20i·7-s + 0.921i·8-s + 0.333·9-s + 0.465·10-s − 1.37i·11-s + 0.409·12-s + (−0.847 + 0.531i)13-s − 0.650·14-s − 0.498i·15-s + 0.213·16-s − 1.22·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$39$$    =    $$3 \cdot 13$$ Sign: $0.847 - 0.531i$ Analytic conductor: $$2.30107$$ Root analytic conductor: $$1.51692$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{39} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 39,\ (\ :3/2),\ 0.847 - 0.531i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.61189 + 0.463421i$$ $$L(\frac12)$$ $$\approx$$ $$1.61189 + 0.463421i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
13 $$1 + (39.7 - 24.8i)T$$
good2 $$1 - 1.52iT - 8T^{2}$$
5 $$1 + 9.65iT - 125T^{2}$$
7 $$1 - 22.3iT - 343T^{2}$$
11 $$1 + 50.3iT - 1.33e3T^{2}$$
17 $$1 + 86.1T + 4.91e3T^{2}$$
19 $$1 + 116. iT - 6.85e3T^{2}$$
23 $$1 + 72T + 1.21e4T^{2}$$
29 $$1 - 14.1T + 2.43e4T^{2}$$
31 $$1 - 196. iT - 2.97e4T^{2}$$
37 $$1 + 154. iT - 5.06e4T^{2}$$
41 $$1 - 265. iT - 6.89e4T^{2}$$
43 $$1 + 211.T + 7.95e4T^{2}$$
47 $$1 - 67.5iT - 1.03e5T^{2}$$
53 $$1 - 686.T + 1.48e5T^{2}$$
59 $$1 + 91.9iT - 2.05e5T^{2}$$
61 $$1 - 329.T + 2.26e5T^{2}$$
67 $$1 + 768. iT - 3.00e5T^{2}$$
71 $$1 + 264. iT - 3.57e5T^{2}$$
73 $$1 - 771. iT - 3.89e5T^{2}$$
79 $$1 - 1.22e3T + 4.93e5T^{2}$$
83 $$1 - 514. iT - 5.71e5T^{2}$$
89 $$1 - 527. iT - 7.04e5T^{2}$$
97 $$1 - 74.2iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$