↑

Properties

Label 2-3e3-3.2-c10-0-3
Degree $2$
Conductor $27$
Sign $-i$
Analytic cond. $17.1546$
Root an. cond. $4.14181$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8.26i·2-s + 955.·4-s + 2.84e3i·5-s + 2.96e3·7-s + 1.63e4i·8-s − 2.34e4·10-s − 5.70e4i·11-s + 1.76e5·13-s + 2.44e4i·14-s + 8.43e5·16-s + 2.37e6i·17-s − 2.78e6·19-s + 2.71e6i·20-s + 4.71e5·22-s + 6.45e6i·23-s + ⋯
L(s)  = 1  + 0.258i·2-s + 0.933·4-s + 0.909i·5-s + 0.176·7-s + 0.499i·8-s − 0.234·10-s − 0.354i·11-s + 0.474·13-s + 0.0455i·14-s + 0.804·16-s + 1.67i·17-s − 1.12·19-s + 0.848i·20-s + 0.0915·22-s + 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-i$
Analytic conductor: \(17.1546\)
Root analytic conductor: \(4.14181\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :5),\ -i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.59000 + 1.59000i\)
\(L(\frac12)\) \(\approx\) \(1.59000 + 1.59000i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 8.26iT - 1.02e3T^{2} \)
5 \( 1 - 2.84e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.96e3T + 2.82e8T^{2} \)
11 \( 1 + 5.70e4iT - 2.59e10T^{2} \)
13 \( 1 - 1.76e5T + 1.37e11T^{2} \)
17 \( 1 - 2.37e6iT - 2.01e12T^{2} \)
19 \( 1 + 2.78e6T + 6.13e12T^{2} \)
23 \( 1 - 6.45e6iT - 4.14e13T^{2} \)
29 \( 1 - 3.39e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.12e7T + 8.19e14T^{2} \)
37 \( 1 - 9.01e7T + 4.80e15T^{2} \)
41 \( 1 + 1.73e8iT - 1.34e16T^{2} \)
43 \( 1 - 4.56e7T + 2.16e16T^{2} \)
47 \( 1 + 1.90e8iT - 5.25e16T^{2} \)
53 \( 1 + 3.24e8iT - 1.74e17T^{2} \)
59 \( 1 + 4.01e8iT - 5.11e17T^{2} \)
61 \( 1 - 6.57e8T + 7.13e17T^{2} \)
67 \( 1 + 2.39e9T + 1.82e18T^{2} \)
71 \( 1 - 2.19e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.96e9T + 4.29e18T^{2} \)
79 \( 1 - 7.14e8T + 9.46e18T^{2} \)
83 \( 1 + 3.95e9iT - 1.55e19T^{2} \)
89 \( 1 + 5.68e9iT - 3.11e19T^{2} \)
97 \( 1 - 8.34e9T + 7.37e19T^{2} \)
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

−15.18671641229164150619509149324, −14.47752307896603179065022730259, −12.76749018869906039949585630329, −11.18444341466619832767840966596, −10.53401866580336478781670875954, −8.400801377214452573785078445942, −6.97913634080153075790242860904, −5.89517241815391602445192784167, −3.45884176110423094779366280872, −1.83074392813609514746081481660, 0.871457218913790271055644022558, 2.48412521665356200993173037815, 4.55759967120560174324048881241, 6.34540470756886862516188759968, 7.919822362842997365752966416618, 9.471612999230756786288659057822, 10.98970419864718575584372228253, 12.09369831178910831055144497109, 13.18361654899534436800949421481, 14.84859027049322452442991577523

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