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Properties

Label 2-3e6-27.4-c1-0-7
Degree $2$
Conductor $729$
Sign $0.597 - 0.802i$
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  + (−0.162 − 0.0591i)2-s + (−1.50 − 1.26i)4-s + (−0.648 + 3.67i)5-s + (2.32 − 1.94i)7-s + (0.343 + 0.594i)8-s + (0.323 − 0.559i)10-s + (−0.432 − 2.45i)11-s + (0.718 − 0.261i)13-s + (−0.492 + 0.179i)14-s + (0.663 + 3.76i)16-s + (−2.31 + 4.00i)17-s + (0.305 + 0.529i)19-s + (5.63 − 4.73i)20-s + (−0.0748 + 0.424i)22-s + (4.99 + 4.19i)23-s + ⋯
L(s)  = 1  + (−0.114 − 0.0418i)2-s + (−0.754 − 0.633i)4-s + (−0.290 + 1.64i)5-s + (0.877 − 0.736i)7-s + (0.121 + 0.210i)8-s + (0.102 − 0.177i)10-s + (−0.130 − 0.739i)11-s + (0.199 − 0.0725i)13-s + (−0.131 + 0.0479i)14-s + (0.165 + 0.940i)16-s + (−0.560 + 0.970i)17-s + (0.0701 + 0.121i)19-s + (1.26 − 1.05i)20-s + (−0.0159 + 0.0904i)22-s + (1.04 + 0.874i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.597 - 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02441 + 0.514481i\)
\(L(\frac12)\) \(\approx\) \(1.02441 + 0.514481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.162 + 0.0591i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.648 - 3.67i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.32 + 1.94i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.432 + 2.45i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.718 + 0.261i)T + (9.95 - 8.35i)T^{2} \)
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 \( 1 + (-4.99 - 4.19i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-6.15 - 2.24i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.01 - 4.21i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.94 - 1.79i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.967 - 5.48i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.848 + 0.711i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + (2.05 - 11.6i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-6.27 + 5.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.13 + 0.414i)T + (51.3 - 43.0i)T^{2} \)
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 \( 1 + (-11.0 - 4.03i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (8.47 + 3.08i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (3.76 + 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.164 + 0.933i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.69568568438080272442989053861, −9.970376216817128886212886040407, −8.667660800131487659596958478575, −8.029918937617561611171533974848, −6.96731752179182844619528212065, −6.21999532291297734426646051907, −5.05763948344203382160652640904, −4.01080249889495994029124424288, −3.01252421001169591663366836455, −1.31625415827193035691615471323, 0.73687240742101349333082322158, 2.39746893023707931713955576590, 4.12841422615817117287684649567, 4.83820437019063841775616375917, 5.28170214283650671869745938781, 6.98579940979834022157365885639, 8.047175967389198419690945955385, 8.620585411419453565883683232859, 9.061781351146001405177264764480, 9.969429275194241288325603482716

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