# Properties

 Label 2-63-7.2-c5-0-9 Degree $2$ Conductor $63$ Sign $0.994 + 0.102i$ 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 + (1.37 − 2.37i)2-s + (12.2 + 21.1i)4-s + (29.1 − 50.5i)5-s + (−21.4 + 127. i)7-s + 155.·8-s + (−80.2 − 138. i)10-s + (8.71 + 15.0i)11-s + 889.·13-s + (274. + 226. i)14-s + (−178. + 308. i)16-s + (513. + 889. i)17-s + (869. − 1.50e3i)19-s + 1.42e3·20-s + 47.8·22-s + (1.96e3 − 3.40e3i)23-s + ⋯
 L(s)  = 1 + (0.242 − 0.420i)2-s + (0.381 + 0.661i)4-s + (0.522 − 0.904i)5-s + (−0.165 + 0.986i)7-s + 0.856·8-s + (−0.253 − 0.439i)10-s + (0.0217 + 0.0376i)11-s + 1.46·13-s + (0.374 + 0.309i)14-s + (−0.173 + 0.301i)16-s + (0.430 + 0.746i)17-s + (0.552 − 0.957i)19-s + 0.797·20-s + 0.0210·22-s + (0.775 − 1.34i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.994 + 0.102i$ Analytic conductor: $$10.1041$$ Root analytic conductor: $$3.17870$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{63} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :5/2),\ 0.994 + 0.102i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.47001 - 0.126996i$$ $$L(\frac12)$$ $$\approx$$ $$2.47001 - 0.126996i$$ $$L(\frac{7}{2})$$ 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 + (21.4 - 127. i)T$$
good2 $$1 + (-1.37 + 2.37i)T + (-16 - 27.7i)T^{2}$$
5 $$1 + (-29.1 + 50.5i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (-8.71 - 15.0i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 - 889.T + 3.71e5T^{2}$$
17 $$1 + (-513. - 889. i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (-869. + 1.50e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-1.96e3 + 3.40e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + 5.63e3T + 2.05e7T^{2}$$
31 $$1 + (-1.54e3 - 2.68e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (2.51e3 - 4.35e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 + 1.83e4T + 1.15e8T^{2}$$
43 $$1 + 1.63e3T + 1.47e8T^{2}$$
47 $$1 + (4.80e3 - 8.31e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (1.16e4 + 2.01e4i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (1.80e3 + 3.12e3i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (1.14e4 - 1.98e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.35e4 + 4.07e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 - 1.59e3T + 1.80e9T^{2}$$
73 $$1 + (2.96e3 + 5.13e3i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (-4.42e4 + 7.66e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 - 9.58e4T + 3.93e9T^{2}$$
89 $$1 + (2.32e4 - 4.02e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + 7.59e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$