# Properties

 Degree $2$ Conductor $63$ Sign $0.189 + 0.981i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.57 + 0.910i)2-s + (−2.34 + 4.05i)4-s + (−7.54 − 13.0i)5-s + (16.2 − 8.80i)7-s − 23.0i·8-s + (23.7 + 13.7i)10-s + (−8.56 − 4.94i)11-s − 67.8i·13-s + (−17.6 + 28.7i)14-s + (2.27 + 3.93i)16-s + (35.0 − 60.7i)17-s + (−53.2 + 30.7i)19-s + 70.7·20-s + 18.0·22-s + (−113. + 65.7i)23-s + ⋯
 L(s)  = 1 + (−0.557 + 0.321i)2-s + (−0.292 + 0.507i)4-s + (−0.674 − 1.16i)5-s + (0.879 − 0.475i)7-s − 1.02i·8-s + (0.752 + 0.434i)10-s + (−0.234 − 0.135i)11-s − 1.44i·13-s + (−0.337 + 0.548i)14-s + (0.0355 + 0.0615i)16-s + (0.500 − 0.866i)17-s + (−0.642 + 0.371i)19-s + 0.790·20-s + 0.174·22-s + (−1.03 + 0.596i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.189 + 0.981i$ Motivic weight: $$3$$ Character: $\chi_{63} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.189 + 0.981i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.558583 - 0.461092i$$ $$L(\frac12)$$ $$\approx$$ $$0.558583 - 0.461092i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (-16.2 + 8.80i)T$$
good2 $$1 + (1.57 - 0.910i)T + (4 - 6.92i)T^{2}$$
5 $$1 + (7.54 + 13.0i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (8.56 + 4.94i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 67.8iT - 2.19e3T^{2}$$
17 $$1 + (-35.0 + 60.7i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (53.2 - 30.7i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (113. - 65.7i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 158. iT - 2.43e4T^{2}$$
31 $$1 + (66.2 + 38.2i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (174. + 301. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 138.T + 6.89e4T^{2}$$
43 $$1 - 539.T + 7.95e4T^{2}$$
47 $$1 + (-111. - 193. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (459. + 265. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-271. + 470. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (116. - 67.0i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (160. - 277. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 416. iT - 3.57e5T^{2}$$
73 $$1 + (-472. - 272. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-161. - 279. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 885.T + 5.71e5T^{2}$$
89 $$1 + (-812. - 1.40e3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 739. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$