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Properties

Label 2-3e3-3.2-c6-0-2
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  + 5.10i·2-s + 37.9·4-s + 60.5i·5-s + 176.·7-s + 520. i·8-s − 309.·10-s + 2.31e3i·11-s − 1.55e3·13-s + 902. i·14-s − 228.·16-s − 2.11e3i·17-s + 9.45e3·19-s + 2.29e3i·20-s − 1.18e4·22-s − 1.77e4i·23-s + ⋯
L(s)  = 1  + 0.638i·2-s + 0.592·4-s + 0.484i·5-s + 0.515·7-s + 1.01i·8-s − 0.309·10-s + 1.74i·11-s − 0.707·13-s + 0.328i·14-s − 0.0556·16-s − 0.430i·17-s + 1.37·19-s + 0.287i·20-s − 1.11·22-s − 1.46i·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

\(L(\frac{7}{2})\) \(\approx\) \(1.32877 + 1.32877i\)
\(L(\frac12)\) \(\approx\) \(1.32877 + 1.32877i\)
\(L(4)\) 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 \( 1 - 5.10iT - 64T^{2} \)
5 \( 1 - 60.5iT - 1.56e4T^{2} \)
7 \( 1 - 176.T + 1.17e5T^{2} \)
11 \( 1 - 2.31e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.55e3T + 4.82e6T^{2} \)
17 \( 1 + 2.11e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.45e3T + 4.70e7T^{2} \)
23 \( 1 + 1.77e4iT - 1.48e8T^{2} \)
29 \( 1 + 3.15e4iT - 5.94e8T^{2} \)
31 \( 1 + 5.04e4T + 8.87e8T^{2} \)
37 \( 1 - 4.26e4T + 2.56e9T^{2} \)
41 \( 1 - 2.83e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.91e4T + 6.32e9T^{2} \)
47 \( 1 + 6.15e4iT - 1.07e10T^{2} \)
53 \( 1 + 7.77e4iT - 2.21e10T^{2} \)
59 \( 1 + 3.30e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.72e4T + 5.15e10T^{2} \)
67 \( 1 - 3.81e5T + 9.04e10T^{2} \)
71 \( 1 - 1.27e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.77e5T + 1.51e11T^{2} \)
79 \( 1 - 7.86e4T + 2.43e11T^{2} \)
83 \( 1 - 1.18e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.10e6T + 8.32e11T^{2} \)
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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

−16.24970639598581000361995025001, −14.97508835547711023571448370124, −14.43631531137966918121246103248, −12.43671431520779429291271824882, −11.23807923422349688427440998717, −9.778400741353619927688880912671, −7.74030963047841981072430607430, −6.84420842384134127255055409448, −4.98452367959994773267931509316, −2.30730205666125662886344636723, 1.21002099334173874007745561147, 3.28382371371266602177277263939, 5.56116539722861916486513866196, 7.51096003738820544113419519334, 9.169973564265543068494978770141, 10.82307789486735683815727368681, 11.68850162477862538934367008234, 12.97597714773127252939315473097, 14.38292029036137652378662797048, 15.92968926219862522498187056591

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