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Properties

Label 2-3e6-27.22-c1-0-7
Degree $2$
Conductor $729$
Sign $0.116 - 0.993i$
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  + (−1.50 + 1.26i)2-s + (0.326 − 1.85i)4-s + (−3.47 + 1.26i)5-s + (−0.407 − 2.31i)7-s + (−0.118 − 0.205i)8-s + (3.64 − 6.31i)10-s + (2.04 + 0.745i)11-s + (−3.61 − 3.03i)13-s + (3.54 + 2.97i)14-s + (3.97 + 1.44i)16-s + (1.46 − 2.54i)17-s + (3.11 + 5.39i)19-s + (1.20 + 6.85i)20-s + (−4.03 + 1.46i)22-s + (0.0901 − 0.511i)23-s + ⋯
L(s)  = 1  + (−1.06 + 0.895i)2-s + (0.163 − 0.925i)4-s + (−1.55 + 0.566i)5-s + (−0.154 − 0.873i)7-s + (−0.0419 − 0.0727i)8-s + (1.15 − 1.99i)10-s + (0.617 + 0.224i)11-s + (−1.00 − 0.840i)13-s + (0.946 + 0.794i)14-s + (0.992 + 0.361i)16-s + (0.355 − 0.616i)17-s + (0.714 + 1.23i)19-s + (0.270 + 1.53i)20-s + (−0.859 + 0.312i)22-s + (0.0187 − 0.106i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367425 + 0.326980i\)
\(L(\frac12)\) \(\approx\) \(0.367425 + 0.326980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 1.26i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (3.47 - 1.26i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.407 + 2.31i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.04 - 0.745i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.61 + 3.03i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.46 + 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 5.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0901 + 0.511i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.67 - 2.24i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.747 - 4.23i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.91 - 1.60i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1 + 0.363i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.0412 + 0.233i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 4.66T + 53T^{2} \)
59 \( 1 + (-12.5 + 4.55i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.638 + 3.61i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.9 - 9.19i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.601 - 1.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.34 - 4.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.80 - 8.22i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.65 - 7.26i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.349 + 0.605i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.65 - 2.42i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.20521597494256122202113624093, −9.827802615794803180327509703354, −8.556698400786951023045955687123, −7.83731234357420520229996884766, −7.23135401094740291530351107998, −6.86359321132378301030494651638, −5.40069490714374318167293787588, −3.98325233066284285627191457485, −3.25451310757297421444214225362, −0.77307404679102834944936487064, 0.59359622012803051811145355827, 2.17268295707162211961773773007, 3.39014910536183546339529289560, 4.46629065031700283669698593334, 5.62234669950247200055937024844, 7.13585524901299626439431500014, 7.86931019622115137161541595672, 8.875383560979489905080447657147, 9.108858999396439335856959573999, 10.08237816345623067091421844490

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