↑

Properties

Label 2-3e6-9.7-c1-0-28
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 no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 1.70i)2-s + (−0.939 − 1.62i)4-s + (−1.85 − 3.20i)5-s + (1.17 − 2.03i)7-s + 0.237·8-s − 7.29·10-s + (1.08 − 1.88i)11-s + (2.35 + 4.08i)13-s + (−2.31 − 4.00i)14-s + (2.11 − 3.66i)16-s − 2.93·17-s − 6.22·19-s + (−3.47 + 6.02i)20-s + (−2.14 − 3.71i)22-s + (−0.259 − 0.449i)23-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)2-s + (−0.469 − 0.813i)4-s + (−0.827 − 1.43i)5-s + (0.443 − 0.768i)7-s + 0.0839·8-s − 2.30·10-s + (0.328 − 0.568i)11-s + (0.654 + 1.13i)13-s + (−0.617 − 1.07i)14-s + (0.528 − 0.915i)16-s − 0.711·17-s − 1.42·19-s + (−0.777 + 1.34i)20-s + (−0.457 − 0.792i)22-s + (−0.0541 − 0.0937i)23-s + ⋯

Functional equation

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

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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93014i\)
\(L(\frac12)\) \(\approx\) \(1.93014i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.984 + 1.70i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.85 + 3.20i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.17 + 2.03i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.08 + 1.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.35 - 4.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.93T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 + (0.259 + 0.449i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.74 + 3.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.15 - 3.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.41T + 37T^{2} \)
41 \( 1 + (1.24 + 2.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.532 - 0.921i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.118 + 0.205i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.66T + 53T^{2} \)
59 \( 1 + (-6.65 - 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.83 + 3.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.14 + 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + 4.68T + 73T^{2} \)
79 \( 1 + (-6.40 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.65 + 9.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.699T + 89T^{2} \)
97 \( 1 + (-3.54 + 6.13i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.34846337425249707838913564466, −8.999955644503859249911016283411, −8.528968858257424237167913513148, −7.49574433022786012543686286107, −6.23332546044501460978532263307, −4.71026177227372019286092634980, −4.36672023006209761849424393482, −3.65168135548236195166880020401, −1.93743128401385330418786249270, −0.837996596324665215434696160837, 2.35938108734026969022796893829, 3.65681817092926623594779448378, 4.52731962386395184010504459026, 5.69299663157403717431613028577, 6.49533918577575983563444705193, 7.09200406985837075296287738815, 8.050865531264579048499637800292, 8.574101097126808858264215392661, 10.17108811436460218076603437598, 10.88461087366536775337403600870

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