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Properties

Label 2-3e6-9.7-c1-0-11
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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29406 + 0.747127i\)
\(L(\frac12)\) \(\approx\) \(1.29406 + 0.747127i\)
\(L(\frac{3}{2})\) 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

−10.69208643340930121438330703277, −9.241885573846309872048818858980, −8.779900298627804750673601280447, −8.055880316302954206645128614632, −7.01998431091000527691547014722, −6.33029287280425891603446972095, −5.29618923388364694651834341513, −3.82069728390806179057060531020, −3.20090265300473896796910375043, −1.30813519505957701374671381589, 1.02574998402616349812769196243, 2.37131374196383314022431500510, 3.48835037150359703236603631012, 4.79931979802666293658113821863, 5.89966664585771753991330664136, 6.71716306124590338692013011834, 7.59086985719870969248976549121, 8.663892871006053536188110054920, 9.749632475432784483770037580589, 10.15476381245971990395813852551

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