# Properties

 Label 2-63-7.2-c5-0-1 Degree $2$ Conductor $63$ Sign $-0.440 + 0.897i$ 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 + (−3.54 + 6.13i)2-s + (−9.12 − 15.8i)4-s + (−41.1 + 71.1i)5-s + (112. + 64.3i)7-s − 97.4·8-s + (−291. − 504. i)10-s + (176. + 305. i)11-s − 885.·13-s + (−793. + 463. i)14-s + (637. − 1.10e3i)16-s + (−212. − 368. i)17-s + (781. − 1.35e3i)19-s + 1.50e3·20-s − 2.49e3·22-s + (1.39e3 − 2.41e3i)23-s + ⋯
 L(s)  = 1 + (−0.626 + 1.08i)2-s + (−0.285 − 0.494i)4-s + (−0.735 + 1.27i)5-s + (0.868 + 0.496i)7-s − 0.538·8-s + (−0.921 − 1.59i)10-s + (0.438 + 0.760i)11-s − 1.45·13-s + (−1.08 + 0.631i)14-s + (0.622 − 1.07i)16-s + (−0.178 − 0.308i)17-s + (0.496 − 0.859i)19-s + 0.839·20-s − 1.09·22-s + (0.549 − 0.951i)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}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.440 + 0.897i$ 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: $$(2,\ 63,\ (\ :5/2),\ -0.440 + 0.897i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.320585 - 0.514224i$$ $$L(\frac12)$$ $$\approx$$ $$0.320585 - 0.514224i$$ $$L(\frac{7}{2})$$ 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 + (-112. - 64.3i)T$$
good2 $$1 + (3.54 - 6.13i)T + (-16 - 27.7i)T^{2}$$
5 $$1 + (41.1 - 71.1i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (-176. - 305. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + 885.T + 3.71e5T^{2}$$
17 $$1 + (212. + 368. i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (-781. + 1.35e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-1.39e3 + 2.41e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + 3.67e3T + 2.05e7T^{2}$$
31 $$1 + (-1.79e3 - 3.11e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (7.14e3 - 1.23e4i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 + 1.43e4T + 1.15e8T^{2}$$
43 $$1 - 7.58e3T + 1.47e8T^{2}$$
47 $$1 + (2.88e3 - 4.99e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (1.26e4 + 2.19e4i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (-2.16e4 - 3.74e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (9.72e3 - 1.68e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-1.47e4 - 2.54e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 - 5.16e4T + 1.80e9T^{2}$$
73 $$1 + (1.89e4 + 3.28e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (2.68e4 - 4.65e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + 8.59e4T + 3.93e9T^{2}$$
89 $$1 + (1.04e4 - 1.80e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + 9.75e4T + 8.58e9T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−14.99588792687239974060142834035, −14.33319416665111161822256248570, −12.19056192415167850125472106804, −11.39420469796954768048672122803, −9.903986808683301795019800814959, −8.568477263583231068181716581836, −7.32880044227816268214429183640, −6.85104592006000596569391144458, −4.94242749797742854067167367827, −2.75374132522611519442951574506, 0.35040597774110444556681253239, 1.61384001157345353117703984267, 3.78453504970872479311715876993, 5.27976850435590782613626765845, 7.65076305140642413957601628306, 8.694732183879807184656204730775, 9.732824789717839361409993371092, 11.10768884600883979281786372526, 11.86805905669546996842840234828, 12.70847034076101533692204859666