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Properties

Label 2-3e6-27.25-c1-0-29
Degree $2$
Conductor $729$
Sign $-0.686 + 0.727i$
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  + (0.300 − 1.70i)2-s + (−0.939 − 0.342i)4-s + (2.65 − 2.22i)5-s + (0.939 − 0.342i)7-s + (0.866 − 1.50i)8-s + (−2.99 − 5.19i)10-s + (−2.65 − 2.22i)11-s + (0.868 + 4.92i)13-s + (−0.300 − 1.70i)14-s + (−3.83 − 3.21i)16-s + (0.5 − 0.866i)19-s + (−3.25 + 1.18i)20-s + (−4.59 + 3.85i)22-s + (6.51 + 2.36i)23-s + (1.21 − 6.89i)25-s + 8.66·26-s + ⋯
L(s)  = 1  + (0.212 − 1.20i)2-s + (−0.469 − 0.171i)4-s + (1.18 − 0.995i)5-s + (0.355 − 0.129i)7-s + (0.306 − 0.530i)8-s + (−0.948 − 1.64i)10-s + (−0.800 − 0.671i)11-s + (0.240 + 1.36i)13-s + (−0.0803 − 0.455i)14-s + (−0.957 − 0.803i)16-s + (0.114 − 0.198i)19-s + (−0.727 + 0.264i)20-s + (−0.979 + 0.822i)22-s + (1.35 + 0.494i)23-s + (0.243 − 1.37i)25-s + 1.69·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.686 + 0.727i$
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: \((2,\ 729,\ (\ :1/2),\ -0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883081 - 2.04721i\)
\(L(\frac12)\) \(\approx\) \(0.883081 - 2.04721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.300 + 1.70i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-2.65 + 2.22i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.939 + 0.342i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.65 + 2.22i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.868 - 4.92i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.51 - 2.36i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.601 - 3.41i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (4.69 + 1.71i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.601 - 3.41i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.25 + 1.18i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-2.65 + 2.22i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.87 - 0.684i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.38 - 7.87i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (5.19 + 9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.20 + 6.82i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.0 - 10.9i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.18700722054856216514550724547, −9.277425558064261322990905750510, −8.876465974020775582839104507821, −7.51873973536084641363063118434, −6.37868333126960020429288477545, −5.25060921636549554577470820489, −4.55973426849479449646016263257, −3.24614602853304289953305633151, −2.04163807764496035973278188877, −1.19475357274974093344576100748, 2.02264235311953138203832965823, 3.03874877494723370325247381901, 4.87262714758576893241418265315, 5.55511068883109287679616244150, 6.25121445900939175952682249904, 7.17643415295584154777043335855, 7.77910628578581401433833208507, 8.797693011515744597603713072820, 9.946108894841672447181167750410, 10.61249588407867013252312480020

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