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Properties

Label 2-3e6-27.7-c1-0-29
Degree $2$
Conductor $729$
Sign $-0.835 - 0.549i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 + 0.300i)2-s + (−0.939 + 0.788i)4-s + (−0.673 − 3.82i)5-s + (−1.67 − 1.40i)7-s + (1.41 − 2.45i)8-s + (1.70 + 2.95i)10-s + (−0.0282 + 0.160i)11-s + (−2.26 − 0.824i)13-s + (1.80 + 0.657i)14-s + (−0.00727 + 0.0412i)16-s + (1.5 + 2.59i)17-s + (−1.79 + 3.11i)19-s + (3.64 + 3.05i)20-s + (−0.0248 − 0.140i)22-s + (−2.17 + 1.82i)23-s + ⋯
L(s)  = 1  + (−0.584 + 0.212i)2-s + (−0.469 + 0.394i)4-s + (−0.301 − 1.70i)5-s + (−0.632 − 0.530i)7-s + (0.501 − 0.868i)8-s + (0.539 + 0.934i)10-s + (−0.00850 + 0.0482i)11-s + (−0.628 − 0.228i)13-s + (0.482 + 0.175i)14-s + (−0.00181 + 0.0103i)16-s + (0.363 + 0.630i)17-s + (−0.412 + 0.714i)19-s + (0.815 + 0.683i)20-s + (−0.00529 − 0.0300i)22-s + (−0.453 + 0.380i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.835 - 0.549i$
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: \(1\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.826 - 0.300i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.673 + 3.82i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.67 + 1.40i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.0282 - 0.160i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.26 + 0.824i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.17 - 1.82i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.31 + 2.29i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.97 - 3.33i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-3.31 - 5.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.45 + 1.98i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.08 - 6.13i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.66 + 4.75i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
59 \( 1 + (-0.889 - 5.04i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.89 + 2.43i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.51 - 2.00i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.65 + 13.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.34 - 7.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.19 + 0.433i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.96 - 2.89i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-3.86 + 6.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.678 - 3.84i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.831578632690819717348629785288, −8.898123839623596293365820264817, −8.219735563431044179513054499588, −7.67841427085957292690811578498, −6.47497593427075594051643364878, −5.18012185744780377033439776125, −4.35801853833068107226980476221, −3.50678411776986203598025252829, −1.32301700358388038993176109676, 0, 2.27231113389203457391997785840, 3.11808074763030258520557945789, 4.47179218328817879593997834614, 5.72036925696362439667627769453, 6.64537503309464185065040199034, 7.40677856036232534594574792995, 8.430189688858237448394770492402, 9.449415743606584889137959980691, 9.988363607939132069746781167456

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