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Properties

Label 2-3e6-9.4-c1-0-5
Degree $2$
Conductor $729$
Sign $-0.500 + 0.866i$
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  + (1.35 + 2.34i)2-s + (−2.65 + 4.60i)4-s + (−0.836 + 1.44i)5-s + (−0.250 − 0.433i)7-s − 8.97·8-s − 4.52·10-s + (−0.958 − 1.66i)11-s + (−1.55 + 2.69i)13-s + (0.677 − 1.17i)14-s + (−6.81 − 11.8i)16-s − 2.66·17-s + 5.79·19-s + (−4.44 − 7.70i)20-s + (2.59 − 4.49i)22-s + (−2.32 + 4.02i)23-s + ⋯
L(s)  = 1  + (0.956 + 1.65i)2-s + (−1.32 + 2.30i)4-s + (−0.373 + 0.647i)5-s + (−0.0946 − 0.163i)7-s − 3.17·8-s − 1.43·10-s + (−0.289 − 0.500i)11-s + (−0.431 + 0.747i)13-s + (0.180 − 0.313i)14-s + (−1.70 − 2.95i)16-s − 0.646·17-s + 1.32·19-s + (−0.994 − 1.72i)20-s + (0.552 − 0.957i)22-s + (−0.484 + 0.838i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.500 + 0.866i$
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: \((2,\ 729,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747064 - 1.29395i\)
\(L(\frac12)\) \(\approx\) \(0.747064 - 1.29395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 2.34i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.836 - 1.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.250 + 0.433i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.958 + 1.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.55 - 2.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
23 \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.30 + 3.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.85T + 37T^{2} \)
41 \( 1 + (5.77 - 10.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.50 - 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.41 - 5.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (1.09 - 1.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.42 + 5.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.24 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.83T + 71T^{2} \)
73 \( 1 - 9.93T + 73T^{2} \)
79 \( 1 + (2.65 + 4.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-3.44 - 5.96i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.33964725410735285942846356982, −9.821682167225457503109078548267, −8.947180449184345585037134674154, −7.88346303467801368355391303010, −7.36188480129248954172188767337, −6.58877199728467454025855925483, −5.74812210581056797440445533222, −4.79712775371049880404924140566, −3.83559741852335127559746048194, −2.93538699010000698528484654027, 0.55846019360582081506188357467, 2.05655809411473315253779068792, 3.09648674052432573159586412908, 4.14559436196107555919402250759, 5.00966455634420113630033064195, 5.61521187765818526932136475598, 7.08667293413932165639663095839, 8.507788649532487056626329566610, 9.232670904468107592741617810586, 10.27086691434518606803163081456

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