# Properties

 Label 2-54-27.5-c8-0-23 Degree $2$ Conductor $54$ Sign $-0.686 - 0.727i$ Analytic cond. $21.9984$ Root an. cond. $4.69024$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (7.27 + 8.66i)2-s + (−64.9 − 48.3i)3-s + (−22.2 + 126. i)4-s + (−403. − 1.10e3i)5-s + (−53.2 − 914. i)6-s + (−619. − 3.51e3i)7-s + (−1.25e3 + 724. i)8-s + (1.88e3 + 6.28e3i)9-s + (6.67e3 − 1.15e4i)10-s + (−7.11e3 + 1.95e4i)11-s + (7.54e3 − 7.11e3i)12-s + (2.16e4 + 1.81e4i)13-s + (2.59e4 − 3.09e4i)14-s + (−2.74e4 + 9.15e4i)15-s + (−1.53e4 − 5.60e3i)16-s + (−6.01e4 − 3.47e4i)17-s + ⋯
 L(s)  = 1 + (0.454 + 0.541i)2-s + (−0.802 − 0.597i)3-s + (−0.0868 + 0.492i)4-s + (−0.645 − 1.77i)5-s + (−0.0411 − 0.705i)6-s + (−0.258 − 1.46i)7-s + (−0.306 + 0.176i)8-s + (0.286 + 0.958i)9-s + (0.667 − 1.15i)10-s + (−0.485 + 1.33i)11-s + (0.363 − 0.343i)12-s + (0.758 + 0.636i)13-s + (0.675 − 0.805i)14-s + (−0.541 + 1.80i)15-s + (−0.234 − 0.0855i)16-s + (−0.720 − 0.415i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$54$$    =    $$2 \cdot 3^{3}$$ Sign: $-0.686 - 0.727i$ Analytic conductor: $$21.9984$$ Root analytic conductor: $$4.69024$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{54} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 54,\ (\ :4),\ -0.686 - 0.727i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.0100124 + 0.0232097i$$ $$L(\frac12)$$ $$\approx$$ $$0.0100124 + 0.0232097i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-7.27 - 8.66i)T$$
3 $$1 + (64.9 + 48.3i)T$$
good5 $$1 + (403. + 1.10e3i)T + (-2.99e5 + 2.51e5i)T^{2}$$
7 $$1 + (619. + 3.51e3i)T + (-5.41e6 + 1.97e6i)T^{2}$$
11 $$1 + (7.11e3 - 1.95e4i)T + (-1.64e8 - 1.37e8i)T^{2}$$
13 $$1 + (-2.16e4 - 1.81e4i)T + (1.41e8 + 8.03e8i)T^{2}$$
17 $$1 + (6.01e4 + 3.47e4i)T + (3.48e9 + 6.04e9i)T^{2}$$
19 $$1 + (7.33e4 + 1.27e5i)T + (-8.49e9 + 1.47e10i)T^{2}$$
23 $$1 + (-3.07e5 - 5.42e4i)T + (7.35e10 + 2.67e10i)T^{2}$$
29 $$1 + (-2.37e5 - 2.83e5i)T + (-8.68e10 + 4.92e11i)T^{2}$$
31 $$1 + (8.74e4 - 4.95e5i)T + (-8.01e11 - 2.91e11i)T^{2}$$
37 $$1 + (1.45e5 - 2.52e5i)T + (-1.75e12 - 3.04e12i)T^{2}$$
41 $$1 + (2.34e5 - 2.78e5i)T + (-1.38e12 - 7.86e12i)T^{2}$$
43 $$1 + (-8.71e5 - 3.17e5i)T + (8.95e12 + 7.51e12i)T^{2}$$
47 $$1 + (3.13e6 - 5.52e5i)T + (2.23e13 - 8.14e12i)T^{2}$$
53 $$1 - 1.14e7iT - 6.22e13T^{2}$$
59 $$1 + (-4.51e6 - 1.24e7i)T + (-1.12e14 + 9.43e13i)T^{2}$$
61 $$1 + (2.27e6 + 1.29e7i)T + (-1.80e14 + 6.55e13i)T^{2}$$
67 $$1 + (2.05e7 + 1.72e7i)T + (7.05e13 + 3.99e14i)T^{2}$$
71 $$1 + (1.61e7 + 9.30e6i)T + (3.22e14 + 5.59e14i)T^{2}$$
73 $$1 + (1.09e7 + 1.88e7i)T + (-4.03e14 + 6.98e14i)T^{2}$$
79 $$1 + (-1.58e7 + 1.32e7i)T + (2.63e14 - 1.49e15i)T^{2}$$
83 $$1 + (-8.05e6 - 9.59e6i)T + (-3.91e14 + 2.21e15i)T^{2}$$
89 $$1 + (6.09e7 - 3.51e7i)T + (1.96e15 - 3.40e15i)T^{2}$$
97 $$1 + (-1.24e8 - 4.51e7i)T + (6.00e15 + 5.03e15i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$