Properties

 Degree $2$ Conductor $63$ Sign $0.440 + 0.897i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (3.91 − 2.26i)2-s + (6.22 − 10.7i)4-s + (0.632 + 1.09i)5-s + (13.4 − 12.7i)7-s − 20.1i·8-s + (4.95 + 2.86i)10-s + (−36.0 − 20.7i)11-s + 85.7i·13-s + (23.5 − 80.3i)14-s + (4.27 + 7.39i)16-s + (−38.8 + 67.3i)17-s + (42.1 − 24.3i)19-s + 15.7·20-s − 188.·22-s + (−78.7 + 45.4i)23-s + ⋯
 L(s)  = 1 + (1.38 − 0.799i)2-s + (0.778 − 1.34i)4-s + (0.0566 + 0.0980i)5-s + (0.723 − 0.690i)7-s − 0.890i·8-s + (0.156 + 0.0905i)10-s + (−0.987 − 0.570i)11-s + 1.82i·13-s + (0.450 − 1.53i)14-s + (0.0667 + 0.115i)16-s + (−0.554 + 0.961i)17-s + (0.509 − 0.293i)19-s + 0.176·20-s − 1.82·22-s + (−0.714 + 0.412i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.440 + 0.897i$ Motivic weight: $$3$$ Character: $\chi_{63} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.440 + 0.897i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.41109 - 1.50178i$$ $$L(\frac12)$$ $$\approx$$ $$2.41109 - 1.50178i$$ $$L(\frac{5}{2})$$ 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 + (-13.4 + 12.7i)T$$
good2 $$1 + (-3.91 + 2.26i)T + (4 - 6.92i)T^{2}$$
5 $$1 + (-0.632 - 1.09i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (36.0 + 20.7i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 - 85.7iT - 2.19e3T^{2}$$
17 $$1 + (38.8 - 67.3i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-42.1 + 24.3i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (78.7 - 45.4i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 151. iT - 2.43e4T^{2}$$
31 $$1 + (-76.3 - 44.0i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (45.2 + 78.4i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 383.T + 6.89e4T^{2}$$
43 $$1 + 227.T + 7.95e4T^{2}$$
47 $$1 + (69.5 + 120. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-289. - 167. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-440. + 762. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-11.3 + 6.57i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-221. + 383. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 341. iT - 3.57e5T^{2}$$
73 $$1 + (798. + 460. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-206. - 357. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 954.T + 5.71e5T^{2}$$
89 $$1 + (-14.8 - 25.7i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 1.19e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$