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Properties

Label 2-3e6-27.25-c1-0-28
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.426 − 2.42i)2-s + (−3.79 − 1.38i)4-s + (2.35 − 1.97i)5-s + (2.49 − 0.909i)7-s + (−2.50 + 4.34i)8-s + (−3.78 − 6.55i)10-s + (2.63 + 2.20i)11-s + (−0.580 − 3.29i)13-s + (−1.13 − 6.43i)14-s + (3.25 + 2.72i)16-s + (1.28 + 2.22i)17-s + (1.04 − 1.81i)19-s + (−11.6 + 4.25i)20-s + (6.46 − 5.42i)22-s + (0.502 + 0.182i)23-s + ⋯
L(s)  = 1  + (0.301 − 1.71i)2-s + (−1.89 − 0.691i)4-s + (1.05 − 0.885i)5-s + (0.944 − 0.343i)7-s + (−0.886 + 1.53i)8-s + (−1.19 − 2.07i)10-s + (0.793 + 0.665i)11-s + (−0.161 − 0.913i)13-s + (−0.303 − 1.71i)14-s + (0.813 + 0.682i)16-s + (0.311 + 0.540i)17-s + (0.240 − 0.416i)19-s + (−2.61 + 0.951i)20-s + (1.37 − 1.15i)22-s + (0.104 + 0.0381i)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: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124736 - 2.14164i\)
\(L(\frac12)\) \(\approx\) \(0.124736 - 2.14164i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.426 + 2.42i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-2.35 + 1.97i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.49 + 0.909i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.63 - 2.20i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.580 + 3.29i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.28 - 2.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.04 + 1.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.502 - 0.182i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.439 - 2.49i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (7.24 + 2.63i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-5.14 - 8.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.848 - 4.81i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.10 - 1.76i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.31 - 1.93i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (1.26 - 1.06i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (13.5 - 4.91i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.02 + 5.78i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-7.40 - 12.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.940 - 1.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.98 + 16.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.689 - 3.90i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.14 + 6.83i)T + (16.8 + 95.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.997641148895346740900295564129, −9.517489726097400129964277875819, −8.706837500096338737967134023445, −7.61951184792734965041092361763, −6.00790292618092974214797917385, −4.96470550367992459532865611278, −4.47632087878740962272066883601, −3.15708211273563896020906645042, −1.79791329237876300298464819695, −1.20313402882653839208630898137, 1.99558050722748540192094610863, 3.66523925382767883091602024044, 4.87742798313754828871429188915, 5.73339752844752652931797490541, 6.33797919167980380649457185932, 7.15969981933833995822319032071, 7.939797930209646930730241660685, 9.051035314638840496683822674877, 9.441012053515798543759941191308, 10.79323213217442126195441933869

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