Properties

Label 2-3e6-243.151-c1-0-8
Degree $2$
Conductor $729$
Sign $0.382 - 0.923i$
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  + (1.39 − 0.387i)2-s + (0.0768 − 0.0463i)4-s + (−1.01 − 0.0395i)5-s + (−0.161 + 0.0252i)7-s + (−1.89 + 2.00i)8-s + (−1.43 + 0.340i)10-s + (−0.568 + 4.16i)11-s + (1.68 + 2.45i)13-s + (−0.215 + 0.0978i)14-s + (−1.94 + 3.69i)16-s + (6.87 + 3.45i)17-s + (0.307 − 5.27i)19-s + (−0.0802 + 0.0442i)20-s + (0.821 + 6.01i)22-s + (2.81 + 7.29i)23-s + ⋯
L(s)  = 1  + (0.984 − 0.274i)2-s + (0.0384 − 0.0231i)4-s + (−0.456 − 0.0176i)5-s + (−0.0611 + 0.00955i)7-s + (−0.669 + 0.710i)8-s + (−0.453 + 0.107i)10-s + (−0.171 + 1.25i)11-s + (0.467 + 0.682i)13-s + (−0.0575 + 0.0261i)14-s + (−0.486 + 0.922i)16-s + (1.66 + 0.837i)17-s + (0.0705 − 1.21i)19-s + (−0.0179 + 0.00989i)20-s + (0.175 + 1.28i)22-s + (0.587 + 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52152 + 1.01659i\)
\(L(\frac12)\) \(\approx\) \(1.52152 + 1.01659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 + 0.387i)T + (1.71 - 1.03i)T^{2} \)
5 \( 1 + (1.01 + 0.0395i)T + (4.98 + 0.387i)T^{2} \)
7 \( 1 + (0.161 - 0.0252i)T + (6.66 - 2.13i)T^{2} \)
11 \( 1 + (0.568 - 4.16i)T + (-10.5 - 2.94i)T^{2} \)
13 \( 1 + (-1.68 - 2.45i)T + (-4.68 + 12.1i)T^{2} \)
17 \( 1 + (-6.87 - 3.45i)T + (10.1 + 13.6i)T^{2} \)
19 \( 1 + (-0.307 + 5.27i)T + (-18.8 - 2.20i)T^{2} \)
23 \( 1 + (-2.81 - 7.29i)T + (-17.0 + 15.4i)T^{2} \)
29 \( 1 + (7.18 - 5.13i)T + (9.38 - 27.4i)T^{2} \)
31 \( 1 + (-1.77 - 2.03i)T + (-4.19 + 30.7i)T^{2} \)
37 \( 1 + (-2.96 + 3.98i)T + (-10.6 - 35.4i)T^{2} \)
41 \( 1 + (2.73 + 10.6i)T + (-35.8 + 19.8i)T^{2} \)
43 \( 1 + (-0.972 + 0.882i)T + (4.16 - 42.7i)T^{2} \)
47 \( 1 + (-2.91 + 3.34i)T + (-6.36 - 46.5i)T^{2} \)
53 \( 1 + (-1.30 - 7.41i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (9.14 + 3.73i)T + (42.1 + 41.3i)T^{2} \)
61 \( 1 + (1.32 + 0.801i)T + (28.4 + 53.9i)T^{2} \)
67 \( 1 + (-5.87 - 4.20i)T + (21.6 + 63.3i)T^{2} \)
71 \( 1 + (-1.31 + 4.37i)T + (-59.3 - 39.0i)T^{2} \)
73 \( 1 + (-3.87 - 0.917i)T + (65.2 + 32.7i)T^{2} \)
79 \( 1 + (-7.95 - 7.79i)T + (1.53 + 78.9i)T^{2} \)
83 \( 1 + (1.44 - 5.60i)T + (-72.6 - 40.0i)T^{2} \)
89 \( 1 + (1.18 + 3.94i)T + (-74.3 + 48.9i)T^{2} \)
97 \( 1 + (-5.41 + 0.210i)T + (96.7 - 7.51i)T^{2} \)
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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

−10.84200833677158942803713159316, −9.623410104325637713886003450827, −8.974816958701404967053824421163, −7.79553169119261579375753691252, −7.10378057057733846017757743154, −5.76460891890493635958046350864, −5.03868904857739685652059526244, −3.99251382032091112328097746318, −3.33699177958311045636544308563, −1.85376810605837827297434089825, 0.71874771632796074597455271387, 3.08088886309032962873278489311, 3.65481000968587599641851496279, 4.82951488644132654615014055388, 5.79616263188837531912883929683, 6.25635289252315335435782096163, 7.72673872844008847060560377860, 8.233597689359697838193374363911, 9.487058809234202230725161627078, 10.19910913804457871739731669769

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