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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

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 \( 1 + 4.05e7T + 7.83e15T^{2} \)
show more
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

−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

Graph of the $Z$-function along the critical line