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Properties

Label 2-3e3-9.5-c6-0-2
Degree $2$
Conductor $27$
Sign $0.634 + 0.772i$
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  + (−6.96 + 4.02i)2-s + (0.352 − 0.610i)4-s + (−80.3 − 46.3i)5-s + (60.0 + 103. i)7-s − 509. i·8-s + 746.·10-s + (1.29e3 − 750. i)11-s + (2.14e3 − 3.71e3i)13-s + (−836. − 482. i)14-s + (2.07e3 + 3.58e3i)16-s + 940. i·17-s − 8.39e3·19-s + (−56.6 + 32.6i)20-s + (−6.03e3 + 1.04e4i)22-s + (−7.82e3 − 4.51e3i)23-s + ⋯
L(s)  = 1  + (−0.870 + 0.502i)2-s + (0.00550 − 0.00954i)4-s + (−0.642 − 0.370i)5-s + (0.174 + 0.303i)7-s − 0.994i·8-s + 0.746·10-s + (0.976 − 0.563i)11-s + (0.975 − 1.68i)13-s + (−0.304 − 0.175i)14-s + (0.505 + 0.875i)16-s + 0.191i·17-s − 1.22·19-s + (−0.00707 + 0.00408i)20-s + (−0.566 + 0.982i)22-s + (−0.642 − 0.371i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.634 + 0.772i$
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: \((2,\ 27,\ (\ :3),\ 0.634 + 0.772i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.625661 - 0.295743i\)
\(L(\frac12)\) \(\approx\) \(0.625661 - 0.295743i\)
\(L(4)\) 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 \( 1 + (6.96 - 4.02i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (80.3 + 46.3i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-60.0 - 103. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-1.29e3 + 750. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (-2.14e3 + 3.71e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 - 940. iT - 2.41e7T^{2} \)
19 \( 1 + 8.39e3T + 4.70e7T^{2} \)
23 \( 1 + (7.82e3 + 4.51e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.16e4 + 6.70e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-6.75e3 + 1.16e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 3.90e4T + 2.56e9T^{2} \)
41 \( 1 + (9.52e4 + 5.50e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (5.77e3 + 9.99e3i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (2.28e4 - 1.31e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 - 1.29e4iT - 2.21e10T^{2} \)
59 \( 1 + (-1.60e5 - 9.29e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-4.87e4 - 8.43e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-4.83e4 + 8.37e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 2.64e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.54e5T + 1.51e11T^{2} \)
79 \( 1 + (-4.31e4 - 7.48e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-5.43e5 + 3.14e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 8.74e5iT - 4.96e11T^{2} \)
97 \( 1 + (-9.51e4 - 1.64e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−16.03415538321382324850489800945, −15.08938947431693378948072051951, −13.24386864062771995977015382018, −12.01760538839179624647978876399, −10.39916580614337785682539785703, −8.668032454075924067085394906403, −8.096041722273367432521065214177, −6.23976921205106576345023995879, −3.85778392557913015040553557465, −0.57002555464125192945446891762, 1.59914540673196016390701056609, 4.18420906900031784797493632436, 6.69075473979220221897318299758, 8.415637873859077878929685716324, 9.599374668998065958729514070548, 11.03974093619845025075487355140, 11.82324850100105910683045426889, 13.87946290799482791164896652686, 14.90534385195644191616395776462, 16.49231248336239710666286856979

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