Properties

Label 2-3e6-81.52-c1-0-1
Degree $2$
Conductor $729$
Sign $-0.656 - 0.754i$
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  + (−0.474 − 0.311i)2-s + (−0.664 − 1.54i)4-s + (1.99 + 2.11i)5-s + (−3.10 + 0.363i)7-s + (−0.362 + 2.05i)8-s + (−0.286 − 1.62i)10-s + (−0.984 − 3.28i)11-s + (0.231 + 3.98i)13-s + (1.58 + 0.797i)14-s + (−1.48 + 1.57i)16-s + (−0.878 − 0.319i)17-s + (−4.55 + 1.65i)19-s + (1.92 − 4.47i)20-s + (−0.558 + 1.86i)22-s + (−6.11 − 0.714i)23-s + ⋯
L(s)  = 1  + (−0.335 − 0.220i)2-s + (−0.332 − 0.770i)4-s + (0.891 + 0.944i)5-s + (−1.17 + 0.137i)7-s + (−0.128 + 0.726i)8-s + (−0.0905 − 0.513i)10-s + (−0.296 − 0.991i)11-s + (0.0643 + 1.10i)13-s + (0.424 + 0.213i)14-s + (−0.372 + 0.394i)16-s + (−0.213 − 0.0775i)17-s + (−1.04 + 0.380i)19-s + (0.431 − 1.00i)20-s + (−0.119 + 0.397i)22-s + (−1.27 − 0.148i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138977 + 0.305202i\)
\(L(\frac12)\) \(\approx\) \(0.138977 + 0.305202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.474 + 0.311i)T + (0.792 + 1.83i)T^{2} \)
5 \( 1 + (-1.99 - 2.11i)T + (-0.290 + 4.99i)T^{2} \)
7 \( 1 + (3.10 - 0.363i)T + (6.81 - 1.61i)T^{2} \)
11 \( 1 + (0.984 + 3.28i)T + (-9.19 + 6.04i)T^{2} \)
13 \( 1 + (-0.231 - 3.98i)T + (-12.9 + 1.50i)T^{2} \)
17 \( 1 + (0.878 + 0.319i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (4.55 - 1.65i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (6.11 + 0.714i)T + (22.3 + 5.30i)T^{2} \)
29 \( 1 + (1.60 - 0.807i)T + (17.3 - 23.2i)T^{2} \)
31 \( 1 + (-0.403 + 0.542i)T + (-8.89 - 29.6i)T^{2} \)
37 \( 1 + (8.73 - 7.32i)T + (6.42 - 36.4i)T^{2} \)
41 \( 1 + (-5.55 + 3.65i)T + (16.2 - 37.6i)T^{2} \)
43 \( 1 + (2.17 + 0.515i)T + (38.4 + 19.2i)T^{2} \)
47 \( 1 + (-1.89 - 2.54i)T + (-13.4 + 45.0i)T^{2} \)
53 \( 1 + (-4.18 - 7.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.44 - 4.81i)T + (-49.2 - 32.4i)T^{2} \)
61 \( 1 + (0.226 - 0.523i)T + (-41.8 - 44.3i)T^{2} \)
67 \( 1 + (12.2 + 6.13i)T + (40.0 + 53.7i)T^{2} \)
71 \( 1 + (-2.31 - 13.1i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.17 - 6.64i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.622 + 0.409i)T + (31.2 + 72.5i)T^{2} \)
83 \( 1 + (3.04 + 1.99i)T + (32.8 + 76.2i)T^{2} \)
89 \( 1 + (-2.27 + 12.9i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (8.58 - 9.09i)T + (-5.64 - 96.8i)T^{2} \)
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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

−10.42540083488478274796626686881, −10.01537978430212273080955128488, −9.165118041439009355396665219081, −8.480640415361937012442407441277, −6.91504332043710161833443237944, −6.15744475481999157258804957643, −5.77323420293387073898557287998, −4.20427008102093014594336722245, −2.88259204429985762957487094267, −1.87652597367548491595887445336, 0.17937301037948930476618196627, 2.18114871453987805255490507940, 3.52370878678486761231383031159, 4.57980058705332868057848164225, 5.66556357527403373385196992884, 6.61091502842425820494107218547, 7.57522639538793139457186196444, 8.477750679980054671169783220443, 9.254759039274796999631213109511, 9.851082816136444863894595812517

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