# Properties

 Degree $2$ Conductor $63$ Sign $-0.577 + 0.816i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.40i·2-s − 21.2·4-s − 18.5·7-s − 71.5i·8-s + 66.7i·11-s − 100. i·14-s + 216.·16-s − 361.·22-s + 125. i·23-s − 125·25-s + 393.·28-s + 69.7i·29-s + 600. i·32-s + 10.5·37-s + 534.·43-s − 1.41e3i·44-s + ⋯
 L(s)  = 1 + 1.91i·2-s − 2.65·4-s − 0.999·7-s − 3.16i·8-s + 1.83i·11-s − 1.91i·14-s + 3.38·16-s − 3.49·22-s + 1.13i·23-s − 25-s + 2.65·28-s + 0.446i·29-s + 3.31i·32-s + 0.0470·37-s + 1.89·43-s − 4.85i·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.577 + 0.816i$ Motivic weight: $$3$$ Character: $\chi_{63} (62, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ -0.577 + 0.816i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.333519 - 0.644309i$$ $$L(\frac12)$$ $$\approx$$ $$0.333519 - 0.644309i$$ $$L(\frac{5}{2})$$ 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 + 18.5T$$
good2 $$1 - 5.40iT - 8T^{2}$$
5 $$1 + 125T^{2}$$
11 $$1 - 66.7iT - 1.33e3T^{2}$$
13 $$1 - 2.19e3T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 - 6.85e3T^{2}$$
23 $$1 - 125. iT - 1.21e4T^{2}$$
29 $$1 - 69.7iT - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 - 10.5T + 5.06e4T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 534.T + 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 65.4iT - 1.48e5T^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 - 2.26e5T^{2}$$
67 $$1 - 740T + 3.00e5T^{2}$$
71 $$1 - 205. iT - 3.57e5T^{2}$$
73 $$1 - 3.89e5T^{2}$$
79 $$1 + 1.38e3T + 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$