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Properties

Label 2-3e6-9.7-c1-0-19
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  + (0.642 − 1.11i)2-s + (0.173 + 0.300i)4-s + (0.223 + 0.386i)5-s + (1.76 − 3.05i)7-s + 3.01·8-s + 0.573·10-s + (−1.39 + 2.40i)11-s + (−1.64 − 2.84i)13-s + (−2.27 − 3.93i)14-s + (1.59 − 2.75i)16-s + 7.03·17-s − 5.18·19-s + (−0.0775 + 0.134i)20-s + (1.78 + 3.09i)22-s + (3.63 + 6.30i)23-s + ⋯
L(s)  = 1  + (0.454 − 0.787i)2-s + (0.0868 + 0.150i)4-s + (0.0998 + 0.172i)5-s + (0.667 − 1.15i)7-s + 1.06·8-s + 0.181·10-s + (−0.419 + 0.725i)11-s + (−0.456 − 0.790i)13-s + (−0.606 − 1.05i)14-s + (0.398 − 0.689i)16-s + 1.70·17-s − 1.18·19-s + (−0.0173 + 0.0300i)20-s + (0.380 + 0.659i)22-s + (0.758 + 1.31i)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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02421 - 1.16867i\)
\(L(\frac12)\) \(\approx\) \(2.02421 - 1.16867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.642 + 1.11i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.223 - 0.386i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.76 + 3.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.39 - 2.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.64 + 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.03T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + (-3.63 - 6.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.80 + 3.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.967 - 1.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.22T + 37T^{2} \)
41 \( 1 + (2.43 + 4.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.87 + 4.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.50 + 2.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.77T + 53T^{2} \)
59 \( 1 + (-1.48 - 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.71 - 8.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.30T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + (-5.95 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.12 - 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 + (5.05 - 8.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40324236031735321070997655530, −9.931013857891255918488055203597, −8.314005470712801678731825579230, −7.56969896090175983149949327449, −7.04409263333432017851667769239, −5.43792992110852336496678327914, −4.56298613049790543596058773064, −3.64940731605354518447549802817, −2.58991793415285694948798626767, −1.28206134066438379642344795401, 1.57585044773283256261531834619, 2.89137451273538796769237969400, 4.58442401604163019246432737048, 5.23166751522657918709639701759, 6.00121395096607597102662754885, 6.84513785720443748805156969872, 7.962572838776888889686761619189, 8.577390586547842847146894301118, 9.579991370689528839309761401169, 10.66206655024826191702902918901

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