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Properties

Label 2-3e3-3.2-c6-0-7
Degree $2$
Conductor $27$
Sign $-i$
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  − 14.1i·2-s − 134.·4-s + 137. i·5-s − 514.·7-s + 1.00e3i·8-s + 1.93e3·10-s − 1.21e3i·11-s − 170.·13-s + 7.26e3i·14-s + 5.47e3·16-s + 1.57e3i·17-s − 1.12e4·19-s − 1.85e4i·20-s − 1.71e4·22-s − 1.05e4i·23-s + ⋯
L(s)  = 1  − 1.76i·2-s − 2.10·4-s + 1.09i·5-s − 1.50·7-s + 1.95i·8-s + 1.93·10-s − 0.915i·11-s − 0.0775·13-s + 2.64i·14-s + 1.33·16-s + 0.320i·17-s − 1.64·19-s − 2.31i·20-s − 1.61·22-s − 0.868i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-i$
Analytic conductor: \(6.21146\)
Root analytic conductor: \(2.49228\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3),\ -i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0902390 + 0.0902390i\)
\(L(\frac12)\) \(\approx\) \(0.0902390 + 0.0902390i\)
\(L(4)\) 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 \( 1 + 14.1iT - 64T^{2} \)
5 \( 1 - 137. iT - 1.56e4T^{2} \)
7 \( 1 + 514.T + 1.17e5T^{2} \)
11 \( 1 + 1.21e3iT - 1.77e6T^{2} \)
13 \( 1 + 170.T + 4.82e6T^{2} \)
17 \( 1 - 1.57e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.12e4T + 4.70e7T^{2} \)
23 \( 1 + 1.05e4iT - 1.48e8T^{2} \)
29 \( 1 - 8.66e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.62e4T + 8.87e8T^{2} \)
37 \( 1 - 2.18e4T + 2.56e9T^{2} \)
41 \( 1 + 3.81e4iT - 4.75e9T^{2} \)
43 \( 1 - 2.38e4T + 6.32e9T^{2} \)
47 \( 1 - 2.66e4iT - 1.07e10T^{2} \)
53 \( 1 + 2.13e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.63e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.11e5T + 5.15e10T^{2} \)
67 \( 1 + 3.81e5T + 9.04e10T^{2} \)
71 \( 1 + 6.75e5iT - 1.28e11T^{2} \)
73 \( 1 + 7.62e5T + 1.51e11T^{2} \)
79 \( 1 - 5.15e5T + 2.43e11T^{2} \)
83 \( 1 - 9.03e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.44e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.16e5T + 8.32e11T^{2} \)
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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

−14.70602077666182556688517006087, −13.35217254499857176157784018278, −12.47181958018302089082748430914, −10.94380486664315432848797359820, −10.29759850575312752134050842379, −8.924972782933606673051086079242, −6.42938884155899717089417501161, −3.71439030626939571547904683655, −2.61148876668330449667207642132, −0.06990772488473806550174332336, 4.42432489056320558551390835928, 5.94911344492520936529687517733, 7.22269592426998206064569776166, 8.774490473079452670235927196588, 9.712757088698401587196219388996, 12.62573745941034360210177811473, 13.28758399739109253286011482215, 14.90006550890958373833857342774, 15.90607050091366008886217608971, 16.67062491574330555249036819393

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