Properties

Label 2-3e6-81.34-c1-0-2
Degree $2$
Conductor $729$
Sign $-0.982 - 0.185i$
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  + (−0.0742 + 0.172i)2-s + (1.34 + 1.42i)4-s + (0.0921 + 1.58i)5-s + (−3.93 − 0.933i)7-s + (−0.698 + 0.254i)8-s + (−0.278 − 0.101i)10-s + (−3.21 − 2.11i)11-s + (1.58 + 0.185i)13-s + (0.452 − 0.608i)14-s + (−0.220 + 3.78i)16-s + (−5.79 + 4.86i)17-s + (1.26 + 1.06i)19-s + (−2.13 + 2.26i)20-s + (0.601 − 0.395i)22-s + (−6.23 + 1.47i)23-s + ⋯
L(s)  = 1  + (−0.0524 + 0.121i)2-s + (0.674 + 0.714i)4-s + (0.0411 + 0.707i)5-s + (−1.48 − 0.352i)7-s + (−0.246 + 0.0898i)8-s + (−0.0881 − 0.0321i)10-s + (−0.968 − 0.636i)11-s + (0.440 + 0.0514i)13-s + (0.121 − 0.162i)14-s + (−0.0551 + 0.946i)16-s + (−1.40 + 1.17i)17-s + (0.291 + 0.244i)19-s + (−0.477 + 0.506i)20-s + (0.128 − 0.0843i)22-s + (−1.30 + 0.308i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.982 - 0.185i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0628697 + 0.672321i\)
\(L(\frac12)\) \(\approx\) \(0.0628697 + 0.672321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0742 - 0.172i)T + (-1.37 - 1.45i)T^{2} \)
5 \( 1 + (-0.0921 - 1.58i)T + (-4.96 + 0.580i)T^{2} \)
7 \( 1 + (3.93 + 0.933i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (3.21 + 2.11i)T + (4.35 + 10.1i)T^{2} \)
13 \( 1 + (-1.58 - 0.185i)T + (12.6 + 2.99i)T^{2} \)
17 \( 1 + (5.79 - 4.86i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-1.26 - 1.06i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (6.23 - 1.47i)T + (20.5 - 10.3i)T^{2} \)
29 \( 1 + (-0.163 - 0.219i)T + (-8.31 + 27.7i)T^{2} \)
31 \( 1 + (-0.530 + 1.77i)T + (-25.9 - 17.0i)T^{2} \)
37 \( 1 + (0.295 + 1.67i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (1.18 + 2.74i)T + (-28.1 + 29.8i)T^{2} \)
43 \( 1 + (4.35 - 2.18i)T + (25.6 - 34.4i)T^{2} \)
47 \( 1 + (-0.0382 - 0.127i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (-4.93 - 8.55i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.47 + 4.25i)T + (23.3 - 54.1i)T^{2} \)
61 \( 1 + (2.83 - 3.00i)T + (-3.54 - 60.8i)T^{2} \)
67 \( 1 + (8.03 - 10.7i)T + (-19.2 - 64.1i)T^{2} \)
71 \( 1 + (2.39 + 0.873i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-5.29 + 1.92i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.20 + 7.44i)T + (-54.2 - 57.4i)T^{2} \)
83 \( 1 + (2.17 - 5.03i)T + (-56.9 - 60.3i)T^{2} \)
89 \( 1 + (-6.33 + 2.30i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.108 - 1.86i)T + (-96.3 - 11.2i)T^{2} \)
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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

−10.62926055240743674645602833090, −10.25495360460530876571251148572, −8.967228537412761163585725731865, −8.131243261878240323350558866355, −7.21017388984799570162904800373, −6.41090142885280605650029069956, −5.92891817669135755247455780026, −3.99130122268989259304090443736, −3.25865333711232081716644538680, −2.32553078268635745945514072592, 0.31359849883679997916701555336, 2.13799321434434452929363359429, 3.06752791646007237549458896294, 4.66007311465896986426882487912, 5.53467989770564277631033888316, 6.50462657362258310480997658113, 7.09962806918876035917245269909, 8.441084973806711745716741899411, 9.374488804959682849350553848185, 9.908015098318636020462498047595

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