# Properties

 Degree $2$ Conductor $63$ Sign $-0.645 - 0.763i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.96 + 3.40i)2-s + (1.35 + 5.01i)3-s + (−3.73 + 6.47i)4-s + (1.21 − 2.10i)5-s + (−14.4 + 14.4i)6-s + (3.5 + 6.06i)7-s + 2.05·8-s + (−23.3 + 13.5i)9-s + 9.56·10-s + (−21.0 − 36.5i)11-s + (−37.5 − 9.98i)12-s + (19.7 − 34.2i)13-s + (−13.7 + 23.8i)14-s + (12.2 + 3.24i)15-s + (33.9 + 58.8i)16-s − 2.07·17-s + ⋯
 L(s)  = 1 + (0.695 + 1.20i)2-s + (0.260 + 0.965i)3-s + (−0.467 + 0.809i)4-s + (0.108 − 0.188i)5-s + (−0.981 + 0.985i)6-s + (0.188 + 0.327i)7-s + 0.0908·8-s + (−0.864 + 0.503i)9-s + 0.302·10-s + (−0.577 − 1.00i)11-s + (−0.903 − 0.240i)12-s + (0.422 − 0.731i)13-s + (−0.262 + 0.455i)14-s + (0.210 + 0.0559i)15-s + (0.530 + 0.918i)16-s − 0.0296·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.645 - 0.763i$ Motivic weight: $$3$$ Character: $\chi_{63} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ -0.645 - 0.763i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.921949 + 1.98659i$$ $$L(\frac12)$$ $$\approx$$ $$0.921949 + 1.98659i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.35 - 5.01i)T$$
7 $$1 + (-3.5 - 6.06i)T$$
good2 $$1 + (-1.96 - 3.40i)T + (-4 + 6.92i)T^{2}$$
5 $$1 + (-1.21 + 2.10i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (21.0 + 36.5i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-19.7 + 34.2i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 + 2.07T + 4.91e3T^{2}$$
19 $$1 - 96.1T + 6.85e3T^{2}$$
23 $$1 + (36.8 - 63.8i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (9.60 + 16.6i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-119. + 207. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + 144.T + 5.06e4T^{2}$$
41 $$1 + (-36.1 + 62.5i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (240. + 416. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (147. + 255. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + 627.T + 1.48e5T^{2}$$
59 $$1 + (74.8 - 129. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-315. - 546. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (2.02 - 3.50i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 798.T + 3.57e5T^{2}$$
73 $$1 - 444.T + 3.89e5T^{2}$$
79 $$1 + (-287. - 498. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (645. + 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 - 750.T + 7.04e5T^{2}$$
97 $$1 + (209. + 363. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$