Properties

Label 2-3e3-1.1-c9-0-11
Degree $2$
Conductor $27$
Sign $-1$
Analytic cond. $13.9059$
Root an. cond. $3.72907$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 32.3·2-s + 533.·4-s − 2.07e3·5-s − 1.05e4·7-s + 692.·8-s − 6.69e4·10-s + 3.16e4·11-s + 4.76e4·13-s − 3.40e5·14-s − 2.50e5·16-s − 3.44e4·17-s + 7.63e5·19-s − 1.10e6·20-s + 1.02e6·22-s − 1.48e6·23-s + 2.33e6·25-s + 1.53e6·26-s − 5.61e6·28-s − 4.80e6·29-s − 7.82e6·31-s − 8.46e6·32-s − 1.11e6·34-s + 2.17e7·35-s + 9.43e6·37-s + 2.46e7·38-s − 1.43e6·40-s − 1.22e7·41-s + ⋯
L(s)  = 1  + 1.42·2-s + 1.04·4-s − 1.48·5-s − 1.65·7-s + 0.0598·8-s − 2.11·10-s + 0.651·11-s + 0.462·13-s − 2.36·14-s − 0.956·16-s − 0.100·17-s + 1.34·19-s − 1.54·20-s + 0.930·22-s − 1.10·23-s + 1.19·25-s + 0.660·26-s − 1.72·28-s − 1.26·29-s − 1.52·31-s − 1.42·32-s − 0.142·34-s + 2.45·35-s + 0.827·37-s + 1.91·38-s − 0.0886·40-s − 0.679·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-1$
Analytic conductor: \(13.9059\)
Root analytic conductor: \(3.72907\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 27,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 32.3T + 512T^{2} \)
5 \( 1 + 2.07e3T + 1.95e6T^{2} \)
7 \( 1 + 1.05e4T + 4.03e7T^{2} \)
11 \( 1 - 3.16e4T + 2.35e9T^{2} \)
13 \( 1 - 4.76e4T + 1.06e10T^{2} \)
17 \( 1 + 3.44e4T + 1.18e11T^{2} \)
19 \( 1 - 7.63e5T + 3.22e11T^{2} \)
23 \( 1 + 1.48e6T + 1.80e12T^{2} \)
29 \( 1 + 4.80e6T + 1.45e13T^{2} \)
31 \( 1 + 7.82e6T + 2.64e13T^{2} \)
37 \( 1 - 9.43e6T + 1.29e14T^{2} \)
41 \( 1 + 1.22e7T + 3.27e14T^{2} \)
43 \( 1 - 3.74e7T + 5.02e14T^{2} \)
47 \( 1 + 1.98e7T + 1.11e15T^{2} \)
53 \( 1 + 5.78e7T + 3.29e15T^{2} \)
59 \( 1 - 2.84e7T + 8.66e15T^{2} \)
61 \( 1 - 4.45e7T + 1.16e16T^{2} \)
67 \( 1 + 1.81e8T + 2.72e16T^{2} \)
71 \( 1 + 1.53e8T + 4.58e16T^{2} \)
73 \( 1 - 1.55e7T + 5.88e16T^{2} \)
79 \( 1 + 3.26e8T + 1.19e17T^{2} \)
83 \( 1 + 5.01e7T + 1.86e17T^{2} \)
89 \( 1 - 1.59e8T + 3.50e17T^{2} \)
97 \( 1 + 1.49e8T + 7.60e17T^{2} \)
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

−14.58263373059015470017301210418, −13.24599156214998428173060506962, −12.30398716258338207386282083466, −11.37923595423464343740378403665, −9.314943640556671546391269863093, −7.29368919273501520822689348860, −5.94139963711553859948138082766, −3.98961922030644275337677923316, −3.30916906172802683919540456871, 0, 3.30916906172802683919540456871, 3.98961922030644275337677923316, 5.94139963711553859948138082766, 7.29368919273501520822689348860, 9.314943640556671546391269863093, 11.37923595423464343740378403665, 12.30398716258338207386282083466, 13.24599156214998428173060506962, 14.58263373059015470017301210418

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