↑

Properties

Label 2-3e6-9.7-c1-0-5
Degree $2$
Conductor $729$
Sign $-0.500 - 0.866i$
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  + (−0.207 + 0.359i)2-s + (0.913 + 1.58i)4-s + (1.10 + 1.91i)5-s + (0.659 − 1.14i)7-s − 1.59·8-s − 0.920·10-s + (−2.60 + 4.51i)11-s + (0.00902 + 0.0156i)13-s + (0.274 + 0.474i)14-s + (−1.49 + 2.59i)16-s + 3.13·17-s + 0.417·19-s + (−2.02 + 3.50i)20-s + (−1.08 − 1.87i)22-s + (−0.517 − 0.895i)23-s + ⋯
L(s)  = 1  + (−0.146 + 0.254i)2-s + (0.456 + 0.791i)4-s + (0.495 + 0.857i)5-s + (0.249 − 0.431i)7-s − 0.562·8-s − 0.291·10-s + (−0.786 + 1.36i)11-s + (0.00250 + 0.00433i)13-s + (0.0732 + 0.126i)14-s + (−0.374 + 0.648i)16-s + 0.759·17-s + 0.0957·19-s + (−0.452 + 0.783i)20-s + (−0.230 − 0.400i)22-s + (−0.107 − 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765752 + 1.32632i\)
\(L(\frac12)\) \(\approx\) \(0.765752 + 1.32632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.207 - 0.359i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.659 + 1.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.60 - 4.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.00902 - 0.0156i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.13T + 17T^{2} \)
19 \( 1 - 0.417T + 19T^{2} \)
23 \( 1 + (0.517 + 0.895i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.90 - 6.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.86 + 3.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
41 \( 1 + (1.83 + 3.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.15 + 7.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.54 - 6.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.30T + 53T^{2} \)
59 \( 1 + (1.85 + 3.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.45 - 5.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.51 - 9.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.08T + 71T^{2} \)
73 \( 1 + 0.546T + 73T^{2} \)
79 \( 1 + (0.244 - 0.423i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.30 + 3.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + (4.97 - 8.60i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.59890005749010599770400801293, −9.961603769294057258363541105338, −8.938407395032632239306444058535, −7.68177851706213392768849558293, −7.39084146263659909615802684278, −6.52046743509210277377643861761, −5.42813255683226241996384009170, −4.15195635158513195590779009062, −2.97784538523869449850642382967, −2.03898575322194421021117425103, 0.818556591338197625627502081573, 2.06440481441846815997981337225, 3.26711684272036390865206722349, 4.98248417187141213418906902750, 5.61539572152523417338614023919, 6.23120845690974327341577306335, 7.68408070557342325216957612956, 8.527826892028473687861122579604, 9.372226550861473364355601563529, 10.03591865452686531535208303120

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