# Properties

 Label 2-54-3.2-c8-0-8 Degree $2$ Conductor $54$ Sign $i$ 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 + 11.3i·2-s − 128.·4-s + 678. i·5-s − 2.06e3·7-s − 1.44e3i·8-s − 7.68e3·10-s + 6.65e3i·11-s + 8.06e3·13-s − 2.33e4i·14-s + 1.63e4·16-s − 2.15e4i·17-s − 2.26e5·19-s − 8.68e4i·20-s − 7.52e4·22-s − 3.68e5i·23-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.500·4-s + 1.08i·5-s − 0.860·7-s − 0.353i·8-s − 0.768·10-s + 0.454i·11-s + 0.282·13-s − 0.608i·14-s + 0.250·16-s − 0.258i·17-s − 1.73·19-s − 0.543i·20-s − 0.321·22-s − 1.31i·23-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$54$$    =    $$2 \cdot 3^{3}$$ Sign: $i$ Analytic conductor: $$21.9984$$ Root analytic conductor: $$4.69024$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{54} (53, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 54,\ (\ :4),\ i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.0593174 - 0.0593174i$$ $$L(\frac12)$$ $$\approx$$ $$0.0593174 - 0.0593174i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 11.3iT$$
3 $$1$$
good5 $$1 - 678. iT - 3.90e5T^{2}$$
7 $$1 + 2.06e3T + 5.76e6T^{2}$$
11 $$1 - 6.65e3iT - 2.14e8T^{2}$$
13 $$1 - 8.06e3T + 8.15e8T^{2}$$
17 $$1 + 2.15e4iT - 6.97e9T^{2}$$
19 $$1 + 2.26e5T + 1.69e10T^{2}$$
23 $$1 + 3.68e5iT - 7.83e10T^{2}$$
29 $$1 + 9.37e5iT - 5.00e11T^{2}$$
31 $$1 - 8.26e5T + 8.52e11T^{2}$$
37 $$1 - 1.34e6T + 3.51e12T^{2}$$
41 $$1 + 5.19e6iT - 7.98e12T^{2}$$
43 $$1 + 6.14e6T + 1.16e13T^{2}$$
47 $$1 - 5.91e6iT - 2.38e13T^{2}$$
53 $$1 + 7.68e5iT - 6.22e13T^{2}$$
59 $$1 - 4.73e5iT - 1.46e14T^{2}$$
61 $$1 + 1.49e7T + 1.91e14T^{2}$$
67 $$1 + 1.00e7T + 4.06e14T^{2}$$
71 $$1 - 4.54e7iT - 6.45e14T^{2}$$
73 $$1 + 2.32e7T + 8.06e14T^{2}$$
79 $$1 - 1.42e7T + 1.51e15T^{2}$$
83 $$1 - 3.61e7iT - 2.25e15T^{2}$$
89 $$1 - 1.15e8iT - 3.93e15T^{2}$$
97 $$1 + 4.05e7T + 7.83e15T^{2}$$
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$$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

−13.53452531560391858061370678342, −12.42472790942421970277713339427, −10.78695639459430025697913683898, −9.816006207693416329938664046144, −8.362324261417414539426944629797, −6.87077101950411592059378679206, −6.22114443536005335983964391630, −4.24688015296670408978462999963, −2.63939802955244067024854563713, −0.02970082126334810075167608930, 1.44144272606912720024728855532, 3.31480778809900898652660708313, 4.75056911455626054007110796660, 6.27708731810099713176079237022, 8.305236224768396701721861721573, 9.216199875780054088331923826848, 10.41505064699231845020114047792, 11.73559560542652893167290662129, 12.89539181101130305240661428405, 13.36009667554431964317882168602