Properties

Label 2-3e6-729.4-c1-0-13
Degree $2$
Conductor $729$
Sign $0.950 - 0.309i$
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  + (−2.68 − 0.0347i)2-s + (0.113 − 1.72i)3-s + (5.21 + 0.134i)4-s + (−1.58 − 0.486i)5-s + (−0.363 + 4.63i)6-s + (−0.858 + 2.14i)7-s + (−8.63 − 0.335i)8-s + (−2.97 − 0.390i)9-s + (4.25 + 1.36i)10-s + (−0.0532 − 0.0302i)11-s + (0.822 − 8.99i)12-s + (−1.75 − 3.92i)13-s + (2.38 − 5.72i)14-s + (−1.02 + 2.69i)15-s + (12.7 + 0.660i)16-s + (−0.964 − 0.438i)17-s + ⋯
L(s)  = 1  + (−1.89 − 0.0245i)2-s + (0.0653 − 0.997i)3-s + (2.60 + 0.0674i)4-s + (−0.710 − 0.217i)5-s + (−0.148 + 1.89i)6-s + (−0.324 + 0.809i)7-s + (−3.05 − 0.118i)8-s + (−0.991 − 0.130i)9-s + (1.34 + 0.430i)10-s + (−0.0160 − 0.00913i)11-s + (0.237 − 2.59i)12-s + (−0.485 − 1.08i)13-s + (0.636 − 1.52i)14-s + (−0.263 + 0.694i)15-s + (3.19 + 0.165i)16-s + (−0.233 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.950 - 0.309i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.950 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355918 + 0.0564632i\)
\(L(\frac12)\) \(\approx\) \(0.355918 + 0.0564632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.113 + 1.72i)T \)
good2 \( 1 + (2.68 + 0.0347i)T + (1.99 + 0.0517i)T^{2} \)
5 \( 1 + (1.58 + 0.486i)T + (4.14 + 2.80i)T^{2} \)
7 \( 1 + (0.858 - 2.14i)T + (-5.06 - 4.83i)T^{2} \)
11 \( 1 + (0.0532 + 0.0302i)T + (5.62 + 9.45i)T^{2} \)
13 \( 1 + (1.75 + 3.92i)T + (-8.67 + 9.68i)T^{2} \)
17 \( 1 + (0.964 + 0.438i)T + (11.1 + 12.8i)T^{2} \)
19 \( 1 + (-3.64 - 5.31i)T + (-6.84 + 17.7i)T^{2} \)
23 \( 1 + (2.15 - 6.58i)T + (-18.5 - 13.6i)T^{2} \)
29 \( 1 + (-6.47 + 4.14i)T + (12.1 - 26.3i)T^{2} \)
31 \( 1 + (0.464 + 1.70i)T + (-26.7 + 15.6i)T^{2} \)
37 \( 1 + (-8.92 - 1.75i)T + (34.2 + 13.9i)T^{2} \)
41 \( 1 + (2.32 - 0.880i)T + (30.7 - 27.1i)T^{2} \)
43 \( 1 + (7.34 - 3.22i)T + (29.1 - 31.6i)T^{2} \)
47 \( 1 + (0.596 - 2.19i)T + (-40.5 - 23.7i)T^{2} \)
53 \( 1 + (-1.21 + 1.29i)T + (-3.08 - 52.9i)T^{2} \)
59 \( 1 + (-8.95 + 1.51i)T + (55.6 - 19.4i)T^{2} \)
61 \( 1 + (7.34 - 13.5i)T + (-33.1 - 51.1i)T^{2} \)
67 \( 1 + (-5.09 + 2.63i)T + (38.6 - 54.7i)T^{2} \)
71 \( 1 + (-9.76 - 7.56i)T + (17.7 + 68.7i)T^{2} \)
73 \( 1 + (-2.81 + 0.901i)T + (59.3 - 42.4i)T^{2} \)
79 \( 1 + (-2.10 + 11.0i)T + (-73.5 - 28.9i)T^{2} \)
83 \( 1 + (-1.41 - 8.64i)T + (-78.7 + 26.3i)T^{2} \)
89 \( 1 + (-7.85 - 3.21i)T + (63.5 + 62.3i)T^{2} \)
97 \( 1 + (-3.49 + 15.1i)T + (-87.2 - 42.4i)T^{2} \)
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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

−10.08307113513433561566983052167, −9.553239501187972053401338278933, −8.465454896268522506292162592016, −7.932071936640090516234714724177, −7.51619612126338153163578385588, −6.33954337225968429671345389972, −5.62997748002314451939857229690, −3.24431533981957272599588613446, −2.28458765820646249187053875940, −0.934192830852614355250590333566, 0.44211730230924257177381893810, 2.42339117423696079355491314068, 3.57577843623581545056173743400, 4.79893500354523163640456706006, 6.50960325630273913919759970428, 7.05605054104015427567614697449, 8.023033511162112950453016815157, 8.790540450428822006140431767731, 9.538414970354088591281231706184, 10.15462825009762579889281956062

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