↑

Properties

Label 2-3e6-27.13-c1-0-6
Degree $2$
Conductor $729$
Sign $0.116 - 0.993i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.118 + 0.673i)2-s + (1.43 − 0.524i)4-s + (0.802 + 0.673i)5-s + (0.113 + 0.0412i)7-s + (1.20 + 2.09i)8-s + (−0.358 + 0.620i)10-s + (−4.16 + 3.49i)11-s + (−0.794 + 4.50i)13-s + (−0.0143 + 0.0812i)14-s + (1.08 − 0.907i)16-s + (−2.38 + 4.13i)17-s + (0.294 + 0.509i)19-s + (1.50 + 0.549i)20-s + (−2.84 − 2.38i)22-s + (7.32 − 2.66i)23-s + ⋯
L(s)  = 1  + (0.0839 + 0.476i)2-s + (0.719 − 0.262i)4-s + (0.359 + 0.301i)5-s + (0.0428 + 0.0155i)7-s + (0.427 + 0.739i)8-s + (−0.113 + 0.196i)10-s + (−1.25 + 1.05i)11-s + (−0.220 + 1.24i)13-s + (−0.00382 + 0.0217i)14-s + (0.270 − 0.226i)16-s + (−0.579 + 1.00i)17-s + (0.0675 + 0.116i)19-s + (0.337 + 0.122i)20-s + (−0.607 − 0.509i)22-s + (1.52 − 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42251 + 1.26593i\)
\(L(\frac12)\) \(\approx\) \(1.42251 + 1.26593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.118 - 0.673i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.802 - 0.673i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.113 - 0.0412i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (4.16 - 3.49i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.794 - 4.50i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.38 - 4.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.294 - 0.509i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.32 + 2.66i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.880 - 4.99i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-8.23 + 2.99i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.31 + 7.44i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1 - 0.839i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.27 + 0.826i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + (0.0336 + 0.0282i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-9.59 - 3.49i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.322 - 1.83i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.25 + 5.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.11 + 10.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.121 - 0.691i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.17 - 6.67i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (3.42 + 5.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.91 - 5.80i)T + (16.8 - 95.5i)T^{2} \)
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

−10.52490400663501758622133312427, −9.952968839739956487829124766597, −8.767870317041011134806052257765, −7.81784695899641347579127120371, −6.88912371615983400908499040975, −6.43963999550525961659308901203, −5.24733945253350000671158137674, −4.45599997355821570822878007332, −2.67015658190047962158852866762, −1.89084738641495041672478950555, 0.966454563550970335119922011037, 2.71350252133016895899776913971, 3.11320807913761258177775993431, 4.81807969641304416818076977460, 5.61311025050166861941500784449, 6.69697666311433462081808272432, 7.70506024202875819792116362094, 8.337127422122223696163700531666, 9.570146870777693592366127694154, 10.30944256413542174623594352015

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