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Properties

Label 2-3e6-9.4-c1-0-24
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  + (0.342 + 0.592i)2-s + (0.766 − 1.32i)4-s + (0.524 − 0.907i)5-s + (0.0603 + 0.104i)7-s + 2.41·8-s + 0.716·10-s + (−2.71 − 4.70i)11-s + (2.28 − 3.96i)13-s + (−0.0412 + 0.0714i)14-s + (−0.705 − 1.22i)16-s − 4.77·17-s − 0.588·19-s + (−0.802 − 1.39i)20-s + (1.85 − 3.21i)22-s + (−3.89 + 6.75i)23-s + ⋯
L(s)  = 1  + (0.241 + 0.418i)2-s + (0.383 − 0.663i)4-s + (0.234 − 0.405i)5-s + (0.0227 + 0.0394i)7-s + 0.854·8-s + 0.226·10-s + (−0.819 − 1.41i)11-s + (0.634 − 1.09i)13-s + (−0.0110 + 0.0190i)14-s + (−0.176 − 0.305i)16-s − 1.15·17-s − 0.135·19-s + (−0.179 − 0.310i)20-s + (0.396 − 0.686i)22-s + (−0.812 + 1.40i)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: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61084 - 0.930023i\)
\(L(\frac12)\) \(\approx\) \(1.61084 - 0.930023i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.342 - 0.592i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.524 + 0.907i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0603 - 0.104i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.71 + 4.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.28 + 3.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 0.588T + 19T^{2} \)
23 \( 1 + (3.89 - 6.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.53 - 4.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.37 + 7.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 + (-3.77 + 6.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.652 - 1.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.20 - 2.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 + (0.0219 - 0.0380i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.10 - 8.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.929 + 1.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.51T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + (0.351 + 0.608i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.38 - 5.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.85T + 89T^{2} \)
97 \( 1 + (-4.51 - 7.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47897977329856693160831844221, −9.359662105257509111269445592180, −8.422055625162436079175358992673, −7.67368493504814944596115950225, −6.51193904742547096403857936664, −5.64436757580987395793068387061, −5.25440759999469849068678616597, −3.78917527038449246866658169547, −2.43306030462151300093697595388, −0.892610915951764119589964175890, 2.02600398039401250120128338179, 2.65756551495959916273412622332, 4.18995389721147472249168262725, 4.67880319247512367047451907927, 6.49230636812394303184788190884, 6.82407318140989215497537055187, 7.974623660340447879966704177633, 8.725582727241325444270333035048, 9.971894243862450961649068730208, 10.59644277799374358292569103514

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