# Properties

 Degree $2$ Conductor $63$ Sign $0.313 - 0.949i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.82i·2-s + 21.0·5-s + (−11 − 14.8i)7-s + 22.6i·8-s + 59.5i·10-s + 15.5i·11-s + 29.7i·13-s + (42.1 − 31.1i)14-s − 64.0·16-s − 63.2·17-s − 89.3i·19-s − 44·22-s − 77.7i·23-s + 318.·25-s − 84.2·26-s + ⋯
 L(s)  = 1 + 0.999i·2-s + 1.88·5-s + (−0.593 − 0.804i)7-s + 0.999i·8-s + 1.88i·10-s + 0.426i·11-s + 0.635i·13-s + (0.804 − 0.593i)14-s − 1.00·16-s − 0.901·17-s − 1.07i·19-s − 0.426·22-s − 0.705i·23-s + 2.55·25-s − 0.635·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.313 - 0.949i$ Motivic weight: $$3$$ Character: $\chi_{63} (62, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.313 - 0.949i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.51913 + 1.09768i$$ $$L(\frac12)$$ $$\approx$$ $$1.51913 + 1.09768i$$ $$L(\frac{5}{2})$$ 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 + (11 + 14.8i)T$$
good2 $$1 - 2.82iT - 8T^{2}$$
5 $$1 - 21.0T + 125T^{2}$$
11 $$1 - 15.5iT - 1.33e3T^{2}$$
13 $$1 - 29.7iT - 2.19e3T^{2}$$
17 $$1 + 63.2T + 4.91e3T^{2}$$
19 $$1 + 89.3iT - 6.85e3T^{2}$$
23 $$1 + 77.7iT - 1.21e4T^{2}$$
29 $$1 - 125. iT - 2.43e4T^{2}$$
31 $$1 + 238. iT - 2.97e4T^{2}$$
37 $$1 + 184T + 5.06e4T^{2}$$
41 $$1 + 105.T + 6.89e4T^{2}$$
43 $$1 + 190T + 7.95e4T^{2}$$
47 $$1 - 42.1T + 1.03e5T^{2}$$
53 $$1 + 357. iT - 1.48e5T^{2}$$
59 $$1 - 84.2T + 2.05e5T^{2}$$
61 $$1 + 655. iT - 2.26e5T^{2}$$
67 $$1 - 296T + 3.00e5T^{2}$$
71 $$1 - 329. iT - 3.57e5T^{2}$$
73 $$1 - 804. iT - 3.89e5T^{2}$$
79 $$1 - 836T + 4.93e5T^{2}$$
83 $$1 + 1.22e3T + 5.71e5T^{2}$$
89 $$1 - 695.T + 7.04e5T^{2}$$
97 $$1 + 566. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$