↑

Properties

Label 2-3e6-81.67-c1-0-8
Degree $2$
Conductor $729$
Sign $-0.0419 - 0.999i$
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  + (0.695 − 0.457i)2-s + (−0.517 + 1.19i)4-s + (−0.827 + 0.877i)5-s + (1.30 + 0.152i)7-s + (0.478 + 2.71i)8-s + (−0.174 + 0.989i)10-s + (0.623 − 2.08i)11-s + (−0.264 + 4.54i)13-s + (0.978 − 0.491i)14-s + (−0.218 − 0.231i)16-s + (−3.54 + 1.28i)17-s + (−2.50 − 0.911i)19-s + (−0.623 − 1.44i)20-s + (−0.519 − 1.73i)22-s + (−5.99 + 0.700i)23-s + ⋯
L(s)  = 1  + (0.492 − 0.323i)2-s + (−0.258 + 0.599i)4-s + (−0.370 + 0.392i)5-s + (0.493 + 0.0576i)7-s + (0.169 + 0.958i)8-s + (−0.0551 + 0.312i)10-s + (0.187 − 0.627i)11-s + (−0.0734 + 1.26i)13-s + (0.261 − 0.131i)14-s + (−0.0545 − 0.0578i)16-s + (−0.859 + 0.312i)17-s + (−0.574 − 0.209i)19-s + (−0.139 − 0.323i)20-s + (−0.110 − 0.369i)22-s + (−1.24 + 0.146i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.0419 - 0.999i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.0419 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00870 + 1.05190i\)
\(L(\frac12)\) \(\approx\) \(1.00870 + 1.05190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.695 + 0.457i)T + (0.792 - 1.83i)T^{2} \)
5 \( 1 + (0.827 - 0.877i)T + (-0.290 - 4.99i)T^{2} \)
7 \( 1 + (-1.30 - 0.152i)T + (6.81 + 1.61i)T^{2} \)
11 \( 1 + (-0.623 + 2.08i)T + (-9.19 - 6.04i)T^{2} \)
13 \( 1 + (0.264 - 4.54i)T + (-12.9 - 1.50i)T^{2} \)
17 \( 1 + (3.54 - 1.28i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (2.50 + 0.911i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (5.99 - 0.700i)T + (22.3 - 5.30i)T^{2} \)
29 \( 1 + (-3.71 - 1.86i)T + (17.3 + 23.2i)T^{2} \)
31 \( 1 + (-4.45 - 5.98i)T + (-8.89 + 29.6i)T^{2} \)
37 \( 1 + (-7.47 - 6.27i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-4.83 - 3.18i)T + (16.2 + 37.6i)T^{2} \)
43 \( 1 + (4.91 - 1.16i)T + (38.4 - 19.2i)T^{2} \)
47 \( 1 + (-1.79 + 2.40i)T + (-13.4 - 45.0i)T^{2} \)
53 \( 1 + (-6.22 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.12 + 10.4i)T + (-49.2 + 32.4i)T^{2} \)
61 \( 1 + (-4.67 - 10.8i)T + (-41.8 + 44.3i)T^{2} \)
67 \( 1 + (0.741 - 0.372i)T + (40.0 - 53.7i)T^{2} \)
71 \( 1 + (1.25 - 7.14i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.41 + 7.99i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-9.60 + 6.31i)T + (31.2 - 72.5i)T^{2} \)
83 \( 1 + (3.98 - 2.62i)T + (32.8 - 76.2i)T^{2} \)
89 \( 1 + (0.578 + 3.28i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.935 + 0.991i)T + (-5.64 + 96.8i)T^{2} \)
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

−11.00856042004570129805981653486, −9.790225384652340725998453053864, −8.607257453042038628547038328289, −8.278587501699618335856674881179, −7.08410855004970270945184389244, −6.23953798088098791755215096480, −4.82411021070728030434328871789, −4.18016576383905361952026543446, −3.17398745527377890888558003059, −1.96383750109103358050906573771, 0.64647726221011894895395295102, 2.33080884126237580688750983824, 4.17196133226980041646326970926, 4.53616840891594063095797372340, 5.68817310406926715153178663122, 6.41504073177949743462884840016, 7.62031539768784577518224895796, 8.307001338599810633387782150833, 9.393463849811394277695250070985, 10.18637546305125188518672280856

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