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Properties

Label 2-3e6-9.4-c1-0-10
Degree $2$
Conductor $729$
Sign $0.5 - 0.866i$
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  + (0.207 + 0.359i)2-s + (0.913 − 1.58i)4-s + (−1.10 + 1.91i)5-s + (0.659 + 1.14i)7-s + 1.59·8-s − 0.920·10-s + (2.60 + 4.51i)11-s + (0.00902 − 0.0156i)13-s + (−0.274 + 0.474i)14-s + (−1.49 − 2.59i)16-s − 3.13·17-s + 0.417·19-s + (2.02 + 3.50i)20-s + (−1.08 + 1.87i)22-s + (0.517 − 0.895i)23-s + ⋯
L(s)  = 1  + (0.146 + 0.254i)2-s + (0.456 − 0.791i)4-s + (−0.495 + 0.857i)5-s + (0.249 + 0.431i)7-s + 0.562·8-s − 0.291·10-s + (0.786 + 1.36i)11-s + (0.00250 − 0.00433i)13-s + (−0.0732 + 0.126i)14-s + (−0.374 − 0.648i)16-s − 0.759·17-s + 0.0957·19-s + (0.452 + 0.783i)20-s + (−0.230 + 0.400i)22-s + (0.107 − 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.5 - 0.866i$
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: \((2,\ 729,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51943 + 0.877246i\)
\(L(\frac12)\) \(\approx\) \(1.51943 + 0.877246i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 - 0.359i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.10 - 1.91i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.659 - 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.60 - 4.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.00902 + 0.0156i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 - 0.417T + 19T^{2} \)
23 \( 1 + (-0.517 + 0.895i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.90 - 6.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.86 - 3.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
41 \( 1 + (-1.83 + 3.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.15 - 7.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.54 - 6.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.30T + 53T^{2} \)
59 \( 1 + (-1.85 + 3.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.45 + 5.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.51 + 9.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 + 0.546T + 73T^{2} \)
79 \( 1 + (0.244 + 0.423i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.30 + 3.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + (4.97 + 8.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.70489032839915050723185866519, −9.728884522265365264897247460505, −8.945896919982508547534742006332, −7.61508943466669773993929393038, −6.92828468549285061431627344546, −6.35327455548472327451285745488, −5.12027780206802657918276976746, −4.26107663600288145524046492480, −2.79064842152111645317478439527, −1.62941111586927446646497615605, 0.942077806082286805687160254170, 2.58996885623054026515537616075, 3.90618598406839677672918116741, 4.35500284011603084061215121581, 5.80440640999260606844842058324, 6.82472078853620657370317940369, 7.81913107369866811151029452869, 8.474588929997743560379054485414, 9.148393243024875325243673631237, 10.50405198072719088568369464353

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