# Properties

 Label 2-63-7.4-c5-0-4 Degree $2$ Conductor $63$ Sign $-0.984 + 0.174i$ 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 + (5.11 + 8.85i)2-s + (−36.2 + 62.8i)4-s + (11.8 + 20.5i)5-s + (30.6 + 125. i)7-s − 414.·8-s + (−121. + 210. i)10-s + (232. − 403. i)11-s − 1.01e3·13-s + (−958. + 915. i)14-s + (−957. − 1.65e3i)16-s + (280. − 486. i)17-s + (693. + 1.20e3i)19-s − 1.72e3·20-s + 4.75e3·22-s + (2.05e3 + 3.56e3i)23-s + ⋯
 L(s)  = 1 + (0.903 + 1.56i)2-s + (−1.13 + 1.96i)4-s + (0.212 + 0.367i)5-s + (0.236 + 0.971i)7-s − 2.28·8-s + (−0.383 + 0.665i)10-s + (0.580 − 1.00i)11-s − 1.67·13-s + (−1.30 + 1.24i)14-s + (−0.935 − 1.61i)16-s + (0.235 − 0.408i)17-s + (0.440 + 0.763i)19-s − 0.963·20-s + 2.09·22-s + (0.810 + 1.40i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.984 + 0.174i$ 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: $$(2,\ 63,\ (\ :5/2),\ -0.984 + 0.174i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.212439 - 2.41369i$$ $$L(\frac12)$$ $$\approx$$ $$0.212439 - 2.41369i$$ $$L(\frac{7}{2})$$ 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 + (-30.6 - 125. i)T$$
good2 $$1 + (-5.11 - 8.85i)T + (-16 + 27.7i)T^{2}$$
5 $$1 + (-11.8 - 20.5i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (-232. + 403. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 1.01e3T + 3.71e5T^{2}$$
17 $$1 + (-280. + 486. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-693. - 1.20e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-2.05e3 - 3.56e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 - 2.38e3T + 2.05e7T^{2}$$
31 $$1 + (1.47e3 - 2.55e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-4.95e3 - 8.58e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 - 4.47e3T + 1.15e8T^{2}$$
43 $$1 - 5.18e3T + 1.47e8T^{2}$$
47 $$1 + (1.56e3 + 2.70e3i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-570. + 988. i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-1.37e4 + 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-1.05e4 - 1.82e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-2.77e4 + 4.81e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 - 6.07e3T + 1.80e9T^{2}$$
73 $$1 + (-8.38e3 + 1.45e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-2.42e3 - 4.19e3i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 6.01e4T + 3.93e9T^{2}$$
89 $$1 + (3.12e4 + 5.41e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + 6.36e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$