↑

Properties

Label 2-3e6-27.13-c1-0-30
Degree $2$
Conductor $729$
Sign $-0.993 - 0.116i$
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  + (0.0300 + 0.170i)2-s + (1.85 − 0.673i)4-s + (−2.86 − 2.40i)5-s + (−2.84 − 1.03i)7-s + (0.343 + 0.594i)8-s + (0.323 − 0.559i)10-s + (−1.90 + 1.60i)11-s + (−0.132 + 0.753i)13-s + (0.0910 − 0.516i)14-s + (2.92 − 2.45i)16-s + (−2.31 + 4.00i)17-s + (0.305 + 0.529i)19-s + (−6.91 − 2.51i)20-s + (−0.330 − 0.277i)22-s + (−6.13 + 2.23i)23-s + ⋯
L(s)  = 1  + (0.0212 + 0.120i)2-s + (0.925 − 0.336i)4-s + (−1.28 − 1.07i)5-s + (−1.07 − 0.391i)7-s + (0.121 + 0.210i)8-s + (0.102 − 0.177i)10-s + (−0.575 + 0.482i)11-s + (−0.0368 + 0.208i)13-s + (0.0243 − 0.138i)14-s + (0.731 − 0.614i)16-s + (−0.560 + 0.970i)17-s + (0.0701 + 0.121i)19-s + (−1.54 − 0.562i)20-s + (−0.0703 − 0.0590i)22-s + (−1.27 + 0.465i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.993 - 0.116i$
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: \((2,\ 729,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0151093 + 0.259417i\)
\(L(\frac12)\) \(\approx\) \(0.0151093 + 0.259417i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0300 - 0.170i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (2.86 + 2.40i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.84 + 1.03i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.90 - 1.60i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.132 - 0.753i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.31 - 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.305 - 0.529i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.13 - 2.23i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.13 + 6.45i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.15 - 2.24i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.913 + 5.18i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.26 + 3.58i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.04 + 0.378i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + (9.07 + 7.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.69 + 2.80i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.210 - 1.19i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.45 + 4.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 + 3.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.04 + 11.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.56 - 8.87i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (3.76 + 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.726 - 0.609i)T + (16.8 - 95.5i)T^{2} \)
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

−10.04865187376510803649149408683, −9.070642041370740357088405122943, −7.990426988817516971589820277457, −7.45034794592814035355060386283, −6.47535412340687073546986428867, −5.51397813814039058029563747474, −4.30031986562471751465470205455, −3.48402520877320597201429315941, −1.88913286529967702402153895365, −0.12012016681893575082245411132, 2.57359717610009274468463127718, 3.13422542463803525097879346158, 4.04424404555333537961230938827, 5.75269798916669007351052332490, 6.65715841420751562261648053458, 7.31514557107655790282168384618, 7.977122115163749739467649107801, 9.112604707071176053305445814511, 10.30365484047945026432745356885, 10.92419486764217115274064395761

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