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Properties

Label 2-3e6-27.13-c1-0-29
Degree $2$
Conductor $729$
Sign $-0.686 - 0.727i$
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  + (−0.300 − 1.70i)2-s + (−0.939 + 0.342i)4-s + (−2.65 − 2.22i)5-s + (0.939 + 0.342i)7-s + (−0.866 − 1.50i)8-s + (−2.99 + 5.19i)10-s + (2.65 − 2.22i)11-s + (0.868 − 4.92i)13-s + (0.300 − 1.70i)14-s + (−3.83 + 3.21i)16-s + (0.5 + 0.866i)19-s + (3.25 + 1.18i)20-s + (−4.59 − 3.85i)22-s + (−6.51 + 2.36i)23-s + (1.21 + 6.89i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (−0.212 − 1.20i)2-s + (−0.469 + 0.171i)4-s + (−1.18 − 0.995i)5-s + (0.355 + 0.129i)7-s + (−0.306 − 0.530i)8-s + (−0.948 + 1.64i)10-s + (0.800 − 0.671i)11-s + (0.240 − 1.36i)13-s + (0.0803 − 0.455i)14-s + (−0.957 + 0.803i)16-s + (0.114 + 0.198i)19-s + (0.727 + 0.264i)20-s + (−0.979 − 0.822i)22-s + (−1.35 + 0.494i)23-s + (0.243 + 1.37i)25-s − 1.69·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.686 - 0.727i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.686 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330411 + 0.765978i\)
\(L(\frac12)\) \(\approx\) \(0.330411 + 0.765978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.300 + 1.70i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (2.65 + 2.22i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.65 + 2.22i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.868 + 4.92i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.51 - 2.36i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.601 - 3.41i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (4.69 - 1.71i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.601 - 3.41i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (3.25 + 1.18i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (2.65 + 2.22i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.87 + 0.684i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.38 + 7.87i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.19 + 9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.20 + 6.82i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.0 + 10.9i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.00651766546700391743742269593, −9.009199808918370131259393133197, −8.360190269817817939841212416721, −7.60349128906222541354859185430, −6.18872168982613289868357670276, −5.06476211853572887100472274781, −3.86222157574954521277513250126, −3.31501171786304366492844938198, −1.60532324430562213869251787458, −0.46999858994274609726907188939, 2.21918717256222236748963426204, 3.80824384368043512091731142441, 4.52336858015314157221092221211, 6.01793838319561445440966747196, 6.82689875289288719410760460769, 7.29003073623305743116255231135, 8.088164994621008835861132147430, 8.906914236716621049927362324643, 9.923268502601765814856170456217, 11.14629556924025272178272659423

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