Properties

Label 2-3e3-9.5-c6-0-1
Degree $2$
Conductor $27$
Sign $0.613 - 0.789i$
Analytic cond. $6.21146$
Root an. cond. $2.49228$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.66 − 2.11i)2-s + (−23.0 + 39.9i)4-s + (64.9 + 37.5i)5-s + (181. + 313. i)7-s + 465. i·8-s + 317.·10-s + (1.48e3 − 856. i)11-s + (−1.35e3 + 2.35e3i)13-s + (1.32e3 + 765. i)14-s + (−492. − 852. i)16-s + 843. i·17-s + 3.68e3·19-s + (−2.99e3 + 1.73e3i)20-s + (3.61e3 − 6.26e3i)22-s + (−1.66e4 − 9.59e3i)23-s + ⋯
L(s)  = 1  + (0.457 − 0.264i)2-s + (−0.360 + 0.624i)4-s + (0.519 + 0.300i)5-s + (0.527 + 0.914i)7-s + 0.909i·8-s + 0.317·10-s + (1.11 − 0.643i)11-s + (−0.617 + 1.07i)13-s + (0.483 + 0.278i)14-s + (−0.120 − 0.208i)16-s + 0.171i·17-s + 0.536·19-s + (−0.374 + 0.216i)20-s + (0.339 − 0.588i)22-s + (−1.36 − 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(6.21146\)
Root analytic conductor: \(2.49228\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3),\ 0.613 - 0.789i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.81126 + 0.885858i\)
\(L(\frac12)\) \(\approx\) \(1.81126 + 0.885858i\)
\(L(4)\) 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 \( 1 + (-3.66 + 2.11i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (-64.9 - 37.5i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-181. - 313. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-1.48e3 + 856. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (1.35e3 - 2.35e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 - 843. iT - 2.41e7T^{2} \)
19 \( 1 - 3.68e3T + 4.70e7T^{2} \)
23 \( 1 + (1.66e4 + 9.59e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.69e4 + 1.55e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-4.35e3 + 7.53e3i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 3.21e4T + 2.56e9T^{2} \)
41 \( 1 + (-5.17e4 - 2.98e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (-6.08e4 - 1.05e5i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-7.99e4 + 4.61e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + 1.34e5iT - 2.21e10T^{2} \)
59 \( 1 + (5.21e4 + 3.01e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.13e5 + 1.96e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (3.48e4 - 6.03e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 1.85e5iT - 1.28e11T^{2} \)
73 \( 1 - 8.10e4T + 1.51e11T^{2} \)
79 \( 1 + (-2.46e4 - 4.26e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (6.13e5 - 3.54e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 3.64e5iT - 4.96e11T^{2} \)
97 \( 1 + (8.21e4 + 1.42e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−16.32613367036917004594938853663, −14.46572941668267481129938744685, −13.92409288270230222787442613198, −12.22873663697929653419717995196, −11.53020605088262612490026087754, −9.483866376705133729117368354884, −8.239235605186544853623220268360, −6.15088400389975022498167200685, −4.34443480888145116160339273004, −2.36971387825857193495706041068, 1.16753896831638670830425704925, 4.26023277468938473104231859140, 5.65814070043247942877883210245, 7.35763123166585531873947604364, 9.400476223680039666973607763384, 10.42150882433929438203648077820, 12.27441946145724217683449096079, 13.70399346999656923319910745902, 14.36789242015742358510465672840, 15.62466624746175380222122839831

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