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Properties

Label 2-3e6-27.13-c1-0-21
Degree $2$
Conductor $729$
Sign $0.993 + 0.116i$
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  + (0.469 + 2.66i)2-s + (−4.99 + 1.81i)4-s + (−1.28 − 1.07i)5-s + (−0.470 − 0.171i)7-s + (−4.48 − 7.77i)8-s + (2.26 − 3.91i)10-s + (−1.46 + 1.23i)11-s + (0.540 − 3.06i)13-s + (0.235 − 1.33i)14-s + (10.4 − 8.76i)16-s + (−1.33 + 2.30i)17-s + (−2.89 − 5.02i)19-s + (8.35 + 3.04i)20-s + (−3.97 − 3.33i)22-s + (4.36 − 1.58i)23-s + ⋯
L(s)  = 1  + (0.332 + 1.88i)2-s + (−2.49 + 0.909i)4-s + (−0.572 − 0.480i)5-s + (−0.177 − 0.0647i)7-s + (−1.58 − 2.74i)8-s + (0.715 − 1.23i)10-s + (−0.442 + 0.371i)11-s + (0.149 − 0.850i)13-s + (0.0628 − 0.356i)14-s + (2.61 − 2.19i)16-s + (−0.323 + 0.559i)17-s + (−0.664 − 1.15i)19-s + (1.86 + 0.680i)20-s + (−0.847 − 0.710i)22-s + (0.910 − 0.331i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510441 - 0.0297298i\)
\(L(\frac12)\) \(\approx\) \(0.510441 - 0.0297298i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.469 - 2.66i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (1.28 + 1.07i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.470 + 0.171i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.46 - 1.23i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.540 + 3.06i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.33 - 2.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 5.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.36 + 1.58i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.454 + 2.57i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-4.33 + 1.57i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.00 - 11.3i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (6.89 - 5.78i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.42 + 2.33i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + (1.67 + 1.40i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (6.42 + 2.33i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.16 + 12.2i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.41 + 2.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.96 + 8.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.922 - 5.23i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.473 + 2.68i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (5.60 + 9.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.27 - 4.42i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.02471489778814246250755640938, −9.117245876545851417994517402648, −8.170746652477583337139812956937, −7.924507767542573470295650940822, −6.73126507329632105143509135481, −6.16873706621240783267690305344, −4.85947785929481072459246712174, −4.55715643421008597415977885646, −3.19998748740978983453311695798, −0.24513253960925232578859083948, 1.56570047824011246328203762106, 2.84412156533435121045439586665, 3.60064995062583307654787875184, 4.53229356720081113742266332108, 5.54170318960524769958383283387, 6.86242986581253089930798995141, 8.222734976505611699628825005389, 9.002471193016650064926845889008, 9.869334215848996733220889403861, 10.65153267481478973128583119122

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