Properties

Label 2-3e6-1.1-c1-0-4
Degree $2$
Conductor $729$
Sign $1$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.415·2-s − 1.82·4-s − 2.21·5-s − 1.31·7-s − 1.59·8-s − 0.920·10-s + 5.21·11-s − 0.0180·13-s − 0.548·14-s + 2.99·16-s + 3.13·17-s + 0.417·19-s + 4.04·20-s + 2.16·22-s + 1.03·23-s − 0.0929·25-s − 0.00750·26-s + 2.41·28-s + 7.80·29-s + 3.72·31-s + 4.42·32-s + 1.30·34-s + 2.92·35-s + 4.42·37-s + 0.173·38-s + 3.52·40-s + 3.67·41-s + ⋯
L(s)  = 1  + 0.293·2-s − 0.913·4-s − 0.990·5-s − 0.498·7-s − 0.562·8-s − 0.291·10-s + 1.57·11-s − 0.00500·13-s − 0.146·14-s + 0.748·16-s + 0.759·17-s + 0.0957·19-s + 0.905·20-s + 0.461·22-s + 0.215·23-s − 0.0185·25-s − 0.00147·26-s + 0.455·28-s + 1.44·29-s + 0.669·31-s + 0.782·32-s + 0.223·34-s + 0.494·35-s + 0.727·37-s + 0.0281·38-s + 0.556·40-s + 0.573·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.148516633\)
\(L(\frac12)\) \(\approx\) \(1.148516633\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.415T + 2T^{2} \)
5 \( 1 + 2.21T + 5T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 - 5.21T + 11T^{2} \)
13 \( 1 + 0.0180T + 13T^{2} \)
17 \( 1 - 3.13T + 17T^{2} \)
19 \( 1 - 0.417T + 19T^{2} \)
23 \( 1 - 1.03T + 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 - 3.72T + 31T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
41 \( 1 - 3.67T + 41T^{2} \)
43 \( 1 + 8.30T + 43T^{2} \)
47 \( 1 - 7.09T + 47T^{2} \)
53 \( 1 - 1.30T + 53T^{2} \)
59 \( 1 - 3.70T + 59T^{2} \)
61 \( 1 - 6.91T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 6.08T + 71T^{2} \)
73 \( 1 + 0.546T + 73T^{2} \)
79 \( 1 - 0.489T + 79T^{2} \)
83 \( 1 + 4.61T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 - 9.94T + 97T^{2} \)
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

−10.21417941266125996480060688563, −9.500907583006008548126698169118, −8.682255532387812054201551782929, −7.937722243664903161340267425701, −6.82133717186603785214321640258, −5.94414150730195637885814023140, −4.68307687198509270075605446473, −3.94253613764637251408594541676, −3.16646500836920234526378260300, −0.893855383034941114715143574181, 0.893855383034941114715143574181, 3.16646500836920234526378260300, 3.94253613764637251408594541676, 4.68307687198509270075605446473, 5.94414150730195637885814023140, 6.82133717186603785214321640258, 7.937722243664903161340267425701, 8.682255532387812054201551782929, 9.500907583006008548126698169118, 10.21417941266125996480060688563

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