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Properties

Label 2-3e3-3.2-c8-0-3
Degree $2$
Conductor $27$
Sign $-1$
Analytic cond. $10.9992$
Root an. cond. $3.31650$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.4i·2-s − 14·4-s + 427. i·5-s − 679·7-s + 3.97e3i·8-s − 7.01e3·10-s + 1.33e4i·11-s − 3.08e4·13-s − 1.11e4i·14-s − 6.89e4·16-s − 1.28e5i·17-s − 1.38e5·19-s − 5.98e3i·20-s − 2.19e5·22-s + 3.03e5i·23-s + ⋯
L(s)  = 1  + 1.02i·2-s − 0.0546·4-s + 0.683i·5-s − 0.282·7-s + 0.970i·8-s − 0.701·10-s + 0.913i·11-s − 1.07·13-s − 0.290i·14-s − 1.05·16-s − 1.53i·17-s − 1.06·19-s − 0.0373i·20-s − 0.938·22-s + 1.08i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-1$
Analytic conductor: \(10.9992\)
Root analytic conductor: \(3.31650\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.37556i\)
\(L(\frac12)\) \(\approx\) \(1.37556i\)
\(L(5)\) 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 \( 1 - 16.4iT - 256T^{2} \)
5 \( 1 - 427. iT - 3.90e5T^{2} \)
7 \( 1 + 679T + 5.76e6T^{2} \)
11 \( 1 - 1.33e4iT - 2.14e8T^{2} \)
13 \( 1 + 3.08e4T + 8.15e8T^{2} \)
17 \( 1 + 1.28e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.38e5T + 1.69e10T^{2} \)
23 \( 1 - 3.03e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.32e6iT - 5.00e11T^{2} \)
31 \( 1 - 3.52e5T + 8.52e11T^{2} \)
37 \( 1 - 1.18e6T + 3.51e12T^{2} \)
41 \( 1 + 1.09e6iT - 7.98e12T^{2} \)
43 \( 1 - 6.24e6T + 1.16e13T^{2} \)
47 \( 1 + 2.39e3iT - 2.38e13T^{2} \)
53 \( 1 - 1.25e7iT - 6.22e13T^{2} \)
59 \( 1 + 1.05e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.65e7T + 1.91e14T^{2} \)
67 \( 1 - 7.66e6T + 4.06e14T^{2} \)
71 \( 1 + 2.32e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.49e7T + 8.06e14T^{2} \)
79 \( 1 - 4.16e7T + 1.51e15T^{2} \)
83 \( 1 - 4.47e7iT - 2.25e15T^{2} \)
89 \( 1 + 7.40e5iT - 3.93e15T^{2} \)
97 \( 1 + 1.05e8T + 7.83e15T^{2} \)
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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

−15.99404548306247767802434295871, −14.96883344535685330261818307806, −14.16344382070777094656712276797, −12.39494114115470395758272949270, −10.93686406744189949888581782278, −9.387476464449158266887972907222, −7.49975358554839922812457170890, −6.74705960993782528510696168620, −5.02901659207401034809314560333, −2.55351047092598079542655566795, 0.59446233608563724080527499797, 2.39024521247924996975960912689, 4.18085328076651790579320775088, 6.30580835811470775177758640253, 8.359051456261112438527311289079, 9.874790776959160213759389323245, 11.02038796425741451751658987422, 12.37322608397139633357734425430, 13.07345585418015107305737895843, 14.84714222363863103811257621459

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