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Properties

Label 2-3e6-9.4-c1-0-12
Degree $2$
Conductor $729$
Sign $-0.500 - 0.866i$
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  + (0.840 + 1.45i)2-s + (−0.412 + 0.714i)4-s + (−0.564 + 0.978i)5-s + (1.95 + 3.38i)7-s + 1.97·8-s − 1.89·10-s + (−0.935 − 1.61i)11-s + (0.366 − 0.634i)13-s + (−3.28 + 5.69i)14-s + (2.48 + 4.30i)16-s − 1.88·17-s + 2.74·19-s + (−0.466 − 0.807i)20-s + (1.57 − 2.72i)22-s + (−2.91 + 5.04i)23-s + ⋯
L(s)  = 1  + (0.594 + 1.02i)2-s + (−0.206 + 0.357i)4-s + (−0.252 + 0.437i)5-s + (0.738 + 1.27i)7-s + 0.698·8-s − 0.600·10-s + (−0.281 − 0.488i)11-s + (0.101 − 0.175i)13-s + (−0.878 + 1.52i)14-s + (0.621 + 1.07i)16-s − 0.458·17-s + 0.629·19-s + (−0.104 − 0.180i)20-s + (0.335 − 0.580i)22-s + (−0.607 + 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12105 + 1.94172i\)
\(L(\frac12)\) \(\approx\) \(1.12105 + 1.94172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.840 - 1.45i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.564 - 0.978i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.95 - 3.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.935 + 1.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.366 + 0.634i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.88T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + (2.91 - 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.65 + 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.670 - 1.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + (-0.898 + 1.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.51 + 4.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.854 - 1.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + (5.63 - 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.61 + 4.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.944 + 1.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 + (6.17 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.84 + 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + (-0.171 - 0.297i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93437998807569679097784501326, −9.704379798725681358645556059800, −8.674604381819132429429518145631, −7.87136082827041928943958976081, −7.19608947093863384139163548841, −5.98291629937704100442823547816, −5.56511113834177954738954917887, −4.61735208362062770888971489749, −3.30919482628866088477101171307, −1.90400142377967280073431893790, 1.05427664334892031239895136282, 2.28042098896293615860554128233, 3.65550858544952096077893914809, 4.46053455354863217585432862881, 5.00695080772388301576322106759, 6.67339487759441578996994526883, 7.62206671679035968934886524052, 8.241572906237012246965135295472, 9.566566653028948823252843884882, 10.46064118525462827599430014062

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