# Properties

 Degree $2$ Conductor $63$ Sign $0.699 - 0.714i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.799 − 1.38i)2-s + (2.72 + 4.71i)4-s + (−9.14 + 15.8i)5-s + (12.3 + 13.7i)7-s + 21.4·8-s + (14.6 + 25.3i)10-s + (−30.6 − 53.0i)11-s + 32.4·13-s + (28.9 − 6.15i)14-s + (−4.61 + 7.99i)16-s + (40.6 + 70.4i)17-s + (10.4 − 18.1i)19-s − 99.6·20-s − 97.9·22-s + (16.8 − 29.2i)23-s + ⋯
 L(s)  = 1 + (0.282 − 0.489i)2-s + (0.340 + 0.589i)4-s + (−0.818 + 1.41i)5-s + (0.669 + 0.743i)7-s + 0.949·8-s + (0.462 + 0.800i)10-s + (−0.839 − 1.45i)11-s + 0.692·13-s + (0.552 − 0.117i)14-s + (−0.0721 + 0.124i)16-s + (0.580 + 1.00i)17-s + (0.126 − 0.218i)19-s − 1.11·20-s − 0.948·22-s + (0.152 − 0.264i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.699 - 0.714i$ Motivic weight: $$3$$ Character: $\chi_{63} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.699 - 0.714i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.52415 + 0.641179i$$ $$L(\frac12)$$ $$\approx$$ $$1.52415 + 0.641179i$$ $$L(\frac{5}{2})$$ 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 + (-12.3 - 13.7i)T$$
good2 $$1 + (-0.799 + 1.38i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (9.14 - 15.8i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (30.6 + 53.0i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 - 32.4T + 2.19e3T^{2}$$
17 $$1 + (-40.6 - 70.4i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-10.4 + 18.1i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-16.8 + 29.2i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 52.0T + 2.43e4T^{2}$$
31 $$1 + (96.9 + 167. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-133. + 231. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 203.T + 6.89e4T^{2}$$
43 $$1 + 21.9T + 7.95e4T^{2}$$
47 $$1 + (-123. + 214. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (70.4 + 121. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-110. - 191. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (326. - 565. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (302. + 523. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 716.T + 3.57e5T^{2}$$
73 $$1 + (194. + 336. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-144. + 250. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 115.T + 5.71e5T^{2}$$
89 $$1 + (469. - 813. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 120.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$