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Properties

Label 2-3e6-9.4-c1-0-13
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

\(L(1)\) \(\approx\) \(0.465256 - 0.805847i\)
\(L(\frac12)\) \(\approx\) \(0.465256 - 0.805847i\)
\(L(\frac{3}{2})\) 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

−10.01676702502916039390569185680, −9.390880385745564307488710175219, −8.900250697839647840015593165115, −7.84214699962698994371971940296, −6.99150912077190240146831862910, −5.20355942206666272175444418372, −4.27169061822280792640339697245, −3.17410019866352052897629386414, −1.98984389683490643987638514683, −0.915458701338314574673125688764, 1.06777536318680063592018764150, 3.09039465215261436143747562579, 4.86691614514735662465186939671, 5.72326705122126910874263284828, 6.35103573811662681725944761527, 7.35668534850068312015553107656, 7.84905619751262535743874358748, 8.890069775425453903070162574119, 9.630150414402130290953705783691, 10.23867854041901996558943293036

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