↑

Properties

Label 2-3e6-9.4-c1-0-11
Degree $2$
Conductor $729$
Sign $0.5 + 0.866i$
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  + (−0.840 − 1.45i)2-s + (−0.412 + 0.714i)4-s + (0.564 − 0.978i)5-s + (1.95 + 3.38i)7-s − 1.97·8-s − 1.89·10-s + (0.935 + 1.61i)11-s + (0.366 − 0.634i)13-s + (3.28 − 5.69i)14-s + (2.48 + 4.30i)16-s + 1.88·17-s + 2.74·19-s + (0.466 + 0.807i)20-s + (1.57 − 2.72i)22-s + (2.91 − 5.04i)23-s + ⋯
L(s)  = 1  + (−0.594 − 1.02i)2-s + (−0.206 + 0.357i)4-s + (0.252 − 0.437i)5-s + (0.738 + 1.27i)7-s − 0.698·8-s − 0.600·10-s + (0.281 + 0.488i)11-s + (0.101 − 0.175i)13-s + (0.878 − 1.52i)14-s + (0.621 + 1.07i)16-s + 0.458·17-s + 0.629·19-s + (0.104 + 0.180i)20-s + (0.335 − 0.580i)22-s + (0.607 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12684 - 0.650582i\)
\(L(\frac12)\) \(\approx\) \(1.12684 - 0.650582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.840 + 1.45i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.564 + 0.978i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.95 - 3.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.935 - 1.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.366 + 0.634i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.88T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.670 - 1.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + (0.898 - 1.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.51 + 4.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.854 + 1.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + (-5.63 + 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.61 + 4.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.944 + 1.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 + (6.17 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.84 - 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.72T + 89T^{2} \)
97 \( 1 + (-0.171 - 0.297i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.26635452831357455911990453036, −9.349477473847513223242891120768, −8.854265213611907400628314148395, −8.102593594095234096503575474307, −6.72242724291298571330853392978, −5.57415222381042263918390414393, −4.89277301752572464981850662071, −3.25752156138132276717629517140, −2.21782955143084303662489989591, −1.21746995324517052354198969529, 1.05191670232996038524504172276, 2.98433430494727650872659411881, 4.16256198225971190840955519336, 5.44385355048056090781081311688, 6.38148246013224448877930371428, 7.22007931936271794938116116092, 7.76052302938527436023354194117, 8.591691570744689237358875564298, 9.588358822247405588201080828730, 10.33890101445594633389165357019

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