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Properties

Label 2-3e6-9.4-c1-0-0
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  + (−1.22 − 2.12i)2-s + (−2.02 + 3.49i)4-s + (−1.54 + 2.66i)5-s + (1.32 + 2.30i)7-s + 5.01·8-s + 7.57·10-s + (−1.71 − 2.97i)11-s + (1.67 − 2.89i)13-s + (3.26 − 5.65i)14-s + (−2.12 − 3.67i)16-s − 2.57·17-s − 2.09·19-s + (−6.22 − 10.7i)20-s + (−4.22 + 7.31i)22-s + (0.267 − 0.462i)23-s + ⋯
L(s)  = 1  + (−0.868 − 1.50i)2-s + (−1.01 + 1.74i)4-s + (−0.688 + 1.19i)5-s + (0.502 + 0.870i)7-s + 1.77·8-s + 2.39·10-s + (−0.517 − 0.896i)11-s + (0.463 − 0.803i)13-s + (0.873 − 1.51i)14-s + (−0.530 − 0.919i)16-s − 0.623·17-s − 0.481·19-s + (−1.39 − 2.41i)20-s + (−0.899 + 1.55i)22-s + (0.0557 − 0.0965i)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

\(L(1)\) \(\approx\) \(0.0264339 + 0.0457850i\)
\(L(\frac12)\) \(\approx\) \(0.0264339 + 0.0457850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 2.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.54 - 2.66i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.32 - 2.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.67 + 2.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 + (-0.267 + 0.462i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.85 - 6.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (2.44 - 4.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.37 + 2.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.82 + 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (-0.827 + 1.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.18 + 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.93 + 5.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 1.88T + 73T^{2} \)
79 \( 1 + (8.59 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.98 - 3.43i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 + (-5.31 - 9.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.74700256346114366274391841633, −10.25953161299087164491661525844, −8.846311435290605911673949812192, −8.500480352718331933357542857295, −7.62557724166080171521220063172, −6.41470454696421957994750536721, −5.06646772015830321855094530379, −3.47390177572794939662947317479, −3.06024297758643636099376008926, −1.88325628242928916777574349095, 0.03729868617224404585428071870, 1.52510346913649049204251592041, 4.24729880946969635843351842243, 4.67726268583565390398875627132, 5.78908365103558313347307280983, 6.98228661677326253522733375140, 7.50388312649461562911850068243, 8.356874421143987054585771572300, 8.867879543513456684267117768473, 9.753424925186744719474131461228

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