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Properties

Label 2-3e6-27.7-c1-0-23
Degree $2$
Conductor $729$
Sign $0.973 + 0.230i$
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  + (2.30 − 0.837i)2-s + (3.06 − 2.57i)4-s + (0.425 + 2.41i)5-s + (1.53 + 1.28i)7-s + (2.44 − 4.24i)8-s + (3 + 5.19i)10-s + (−0.425 + 2.41i)11-s + (0.939 + 0.342i)13-s + (4.60 + 1.67i)14-s + (0.694 − 3.93i)16-s + (−3.67 − 6.36i)17-s + (0.5 − 0.866i)19-s + (7.50 + 6.29i)20-s + (1.04 + 5.90i)22-s + (1.87 − 1.57i)23-s + ⋯
L(s)  = 1  + (1.62 − 0.592i)2-s + (1.53 − 1.28i)4-s + (0.190 + 1.07i)5-s + (0.579 + 0.485i)7-s + (0.866 − 1.49i)8-s + (0.948 + 1.64i)10-s + (−0.128 + 0.727i)11-s + (0.260 + 0.0948i)13-s + (1.23 + 0.447i)14-s + (0.173 − 0.984i)16-s + (−0.891 − 1.54i)17-s + (0.114 − 0.198i)19-s + (1.67 + 1.40i)20-s + (0.222 + 1.25i)22-s + (0.391 − 0.328i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.98323 - 0.465573i\)
\(L(\frac12)\) \(\approx\) \(3.98323 - 0.465573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 + 0.837i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.425 - 2.41i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.53 - 1.28i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.425 - 2.41i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.67 + 6.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.87 + 1.57i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.60 - 1.67i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.766 - 0.642i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.60 - 1.67i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.91 + 10.8i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.50 + 6.29i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + (-0.425 - 2.41i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.83 - 3.21i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-6.57 - 2.39i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.67 - 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.57 + 2.39i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.5 - 4.18i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.21 - 6.89i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.88121789868166484886950193205, −9.849441536523448296941708774915, −8.768044653431437042758342511296, −7.17261640832311561308172375227, −6.77453919420735252362426572978, −5.53708688488457511249485491775, −4.91258896519257881061236234673, −3.83140295429779543639666534455, −2.70296862158360008463576048151, −2.06588318468725458854096480770, 1.58199364083437925803951477050, 3.27034683558264249916914133659, 4.27069826177107980375610705371, 4.90713236241288122764767020757, 5.83677593534515801758207337585, 6.50077692123928553988980932080, 7.78271354941877223865188570243, 8.342386322842254027494096519090, 9.435438206574459953553451442540, 10.90766557135854523648033265064

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