# Properties

 Label 2-63-7.4-c5-0-5 Degree $2$ Conductor $63$ Sign $-0.693 - 0.720i$ Analytic cond. $10.1041$ Root an. cond. $3.17870$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.19 + 5.53i)2-s + (−4.41 + 7.64i)4-s + (19.3 + 33.5i)5-s + (−87.5 + 95.6i)7-s + 148.·8-s + (−123. + 214. i)10-s + (−288. + 499. i)11-s + 391.·13-s + (−808. − 178. i)14-s + (614. + 1.06e3i)16-s + (−664. + 1.15e3i)17-s + (−471. − 816. i)19-s − 341.·20-s − 3.68e3·22-s + (−816. − 1.41e3i)23-s + ⋯
 L(s)  = 1 + (0.564 + 0.978i)2-s + (−0.137 + 0.238i)4-s + (0.346 + 0.599i)5-s + (−0.674 + 0.737i)7-s + 0.817·8-s + (−0.391 + 0.677i)10-s + (−0.718 + 1.24i)11-s + 0.642·13-s + (−1.10 − 0.243i)14-s + (0.599 + 1.03i)16-s + (−0.557 + 0.966i)17-s + (−0.299 − 0.518i)19-s − 0.191·20-s − 1.62·22-s + (−0.321 − 0.557i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.693 - 0.720i$ Analytic conductor: $$10.1041$$ Root analytic conductor: $$3.17870$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{63} (46, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :5/2),\ -0.693 - 0.720i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.883186 + 2.07660i$$ $$L(\frac12)$$ $$\approx$$ $$0.883186 + 2.07660i$$ $$L(\frac{7}{2})$$ 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 + (87.5 - 95.6i)T$$
good2 $$1 + (-3.19 - 5.53i)T + (-16 + 27.7i)T^{2}$$
5 $$1 + (-19.3 - 33.5i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (288. - 499. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 - 391.T + 3.71e5T^{2}$$
17 $$1 + (664. - 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (471. + 816. i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (816. + 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 - 1.46e3T + 2.05e7T^{2}$$
31 $$1 + (-1.95e3 + 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 - 1.31e4T + 1.15e8T^{2}$$
43 $$1 - 1.47e4T + 1.47e8T^{2}$$
47 $$1 + (3.40e3 + 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (1.00e3 - 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (2.05e4 + 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 3.99e4T + 1.80e9T^{2}$$
73 $$1 + (-2.78e4 + 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-3.15e4 - 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 4.55e4T + 3.93e9T^{2}$$
89 $$1 + (-7.84e3 - 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 3.12e3T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$