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Properties

Label 2-3e6-81.31-c1-0-30
Degree $2$
Conductor $729$
Sign $0.628 - 0.777i$
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  + (−0.971 − 2.25i)2-s + (−2.75 + 2.92i)4-s + (0.232 − 3.99i)5-s + (−1.28 + 0.303i)7-s + (4.64 + 1.69i)8-s + (−9.21 + 3.35i)10-s + (−2.04 + 1.34i)11-s + (−1.30 + 0.152i)13-s + (1.92 + 2.59i)14-s + (−0.237 − 4.08i)16-s + (0.206 + 0.173i)17-s + (1.02 − 0.862i)19-s + (11.0 + 11.6i)20-s + (5.02 + 3.30i)22-s + (−5.82 − 1.37i)23-s + ⋯
L(s)  = 1  + (−0.686 − 1.59i)2-s + (−1.37 + 1.46i)4-s + (0.103 − 1.78i)5-s + (−0.484 + 0.114i)7-s + (1.64 + 0.597i)8-s + (−2.91 + 1.06i)10-s + (−0.617 + 0.406i)11-s + (−0.361 + 0.0422i)13-s + (0.515 + 0.692i)14-s + (−0.0594 − 1.02i)16-s + (0.0501 + 0.0420i)17-s + (0.235 − 0.197i)19-s + (2.46 + 2.61i)20-s + (1.07 + 0.704i)22-s + (−1.21 − 0.287i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146720 + 0.0700348i\)
\(L(\frac12)\) \(\approx\) \(0.146720 + 0.0700348i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.971 + 2.25i)T + (-1.37 + 1.45i)T^{2} \)
5 \( 1 + (-0.232 + 3.99i)T + (-4.96 - 0.580i)T^{2} \)
7 \( 1 + (1.28 - 0.303i)T + (6.25 - 3.14i)T^{2} \)
11 \( 1 + (2.04 - 1.34i)T + (4.35 - 10.1i)T^{2} \)
13 \( 1 + (1.30 - 0.152i)T + (12.6 - 2.99i)T^{2} \)
17 \( 1 + (-0.206 - 0.173i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.02 + 0.862i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (5.82 + 1.37i)T + (20.5 + 10.3i)T^{2} \)
29 \( 1 + (3.30 - 4.44i)T + (-8.31 - 27.7i)T^{2} \)
31 \( 1 + (-0.369 - 1.23i)T + (-25.9 + 17.0i)T^{2} \)
37 \( 1 + (0.00841 - 0.0477i)T + (-34.7 - 12.6i)T^{2} \)
41 \( 1 + (-1.64 + 3.80i)T + (-28.1 - 29.8i)T^{2} \)
43 \( 1 + (-5.59 - 2.80i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-2.38 + 7.95i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (5.79 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.15 - 5.36i)T + (23.3 + 54.1i)T^{2} \)
61 \( 1 + (-2.93 - 3.10i)T + (-3.54 + 60.8i)T^{2} \)
67 \( 1 + (0.791 + 1.06i)T + (-19.2 + 64.1i)T^{2} \)
71 \( 1 + (-7.40 + 2.69i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (8.12 + 2.95i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-2.07 - 4.80i)T + (-54.2 + 57.4i)T^{2} \)
83 \( 1 + (2.24 + 5.21i)T + (-56.9 + 60.3i)T^{2} \)
89 \( 1 + (8.61 + 3.13i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.721 + 12.3i)T + (-96.3 + 11.2i)T^{2} \)
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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

−9.705810560434402198838056978554, −9.048708929191847305081035765340, −8.418675378930964779452468775076, −7.51337731146621195835306873912, −5.75755122463835918971528152115, −4.74745291813144106471595006553, −3.89201845363056976404870654399, −2.51931276242227832229145971782, −1.43342675397763148518355420044, −0.10661665108787417220062582788, 2.54486833574031601228846913914, 3.79786733011712384701634333137, 5.42064841506694239121628508322, 6.18854626766393970398721693887, 6.76919109346512651050774497212, 7.66264752357498462314836637286, 8.076710396873248744557639901391, 9.556786310364658404131457648671, 9.906451945732342460386298808562, 10.76811644810527460139094407464

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