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Properties

Label 2-3e3-3.2-c10-0-6
Degree $2$
Conductor $27$
Sign $i$
Analytic cond. $17.1546$
Root an. cond. $4.14181$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 49.7i·2-s − 1.45e3·4-s + 4.68e3i·5-s + 1.06e4·7-s + 2.14e4i·8-s + 2.33e5·10-s − 2.10e5i·11-s + 4.83e5·13-s − 5.30e5i·14-s − 4.21e5·16-s + 1.44e6i·17-s + 3.79e6·19-s − 6.81e6i·20-s − 1.04e7·22-s − 2.24e6i·23-s + ⋯
L(s)  = 1  − 1.55i·2-s − 1.42·4-s + 1.49i·5-s + 0.633·7-s + 0.655i·8-s + 2.33·10-s − 1.30i·11-s + 1.30·13-s − 0.985i·14-s − 0.401·16-s + 1.01i·17-s + 1.53·19-s − 2.12i·20-s − 2.03·22-s − 0.348i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $i$
Analytic conductor: \(17.1546\)
Root analytic conductor: \(4.14181\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :5),\ i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.41735 - 1.41735i\)
\(L(\frac12)\) \(\approx\) \(1.41735 - 1.41735i\)
\(L(6)\) 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 \( 1 + 49.7iT - 1.02e3T^{2} \)
5 \( 1 - 4.68e3iT - 9.76e6T^{2} \)
7 \( 1 - 1.06e4T + 2.82e8T^{2} \)
11 \( 1 + 2.10e5iT - 2.59e10T^{2} \)
13 \( 1 - 4.83e5T + 1.37e11T^{2} \)
17 \( 1 - 1.44e6iT - 2.01e12T^{2} \)
19 \( 1 - 3.79e6T + 6.13e12T^{2} \)
23 \( 1 + 2.24e6iT - 4.14e13T^{2} \)
29 \( 1 + 3.09e7iT - 4.20e14T^{2} \)
31 \( 1 - 4.03e7T + 8.19e14T^{2} \)
37 \( 1 - 3.67e7T + 4.80e15T^{2} \)
41 \( 1 - 8.25e7iT - 1.34e16T^{2} \)
43 \( 1 - 9.21e7T + 2.16e16T^{2} \)
47 \( 1 + 3.91e7iT - 5.25e16T^{2} \)
53 \( 1 - 1.93e8iT - 1.74e17T^{2} \)
59 \( 1 + 7.23e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.24e9T + 7.13e17T^{2} \)
67 \( 1 - 1.11e9T + 1.82e18T^{2} \)
71 \( 1 - 6.26e8iT - 3.25e18T^{2} \)
73 \( 1 + 3.06e9T + 4.29e18T^{2} \)
79 \( 1 + 2.38e9T + 9.46e18T^{2} \)
83 \( 1 - 2.39e9iT - 1.55e19T^{2} \)
89 \( 1 - 1.61e9iT - 3.11e19T^{2} \)
97 \( 1 + 6.00e9T + 7.37e19T^{2} \)
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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

−14.22871182251751133504014673265, −13.40712495651679999468879058143, −11.52025001037776191137792091025, −11.10300260832820152055625645235, −10.02086967959701365061172757262, −8.244465280821706447885734166397, −6.20170119003762780598239057623, −3.77768840764197986866250387918, −2.74193105047724683001775353050, −1.06554147559498147290541864680, 1.15802681227280316980296019647, 4.59445040862191678690346039976, 5.44196090365503511966479844748, 7.23610993523597911389951284893, 8.394520386380227069603640462058, 9.411416510836944076765251726046, 11.76224743491302357603199810751, 13.20512072743384289606498419453, 14.27227591974015959520668591195, 15.73310146930385433084302810945

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