Properties

Label 2-3e3-3.2-c4-0-4
Degree $2$
Conductor $27$
Sign $-1$
Analytic cond. $2.79098$
Root an. cond. $1.67062$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.34i·2-s − 38·4-s − 14.6i·5-s + 17·7-s + 161. i·8-s − 108·10-s − 161. i·11-s + 95·13-s − 124. i·14-s + 580.·16-s − 308. i·17-s + 209·19-s + 558. i·20-s − 1.18e3·22-s + 867. i·23-s + ⋯
L(s)  = 1  − 1.83i·2-s − 2.37·4-s − 0.587i·5-s + 0.346·7-s + 2.52i·8-s − 1.08·10-s − 1.33i·11-s + 0.562·13-s − 0.637i·14-s + 2.26·16-s − 1.06i·17-s + 0.578·19-s + 1.39i·20-s − 2.45·22-s + 1.63i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-1$
Analytic conductor: \(2.79098\)
Root analytic conductor: \(1.67062\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :2),\ -1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(-1.14781i\)
\(L(\frac12)\) \(\approx\) \(-1.14781i\)
\(L(3)\) 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 \( 1 + 7.34iT - 16T^{2} \)
5 \( 1 + 14.6iT - 625T^{2} \)
7 \( 1 - 17T + 2.40e3T^{2} \)
11 \( 1 + 161. iT - 1.46e4T^{2} \)
13 \( 1 - 95T + 2.85e4T^{2} \)
17 \( 1 + 308. iT - 8.35e4T^{2} \)
19 \( 1 - 209T + 1.30e5T^{2} \)
23 \( 1 - 867. iT - 2.79e5T^{2} \)
29 \( 1 - 323. iT - 7.07e5T^{2} \)
31 \( 1 - 950T + 9.23e5T^{2} \)
37 \( 1 + 1.17e3T + 1.87e6T^{2} \)
41 \( 1 - 2.14e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.43e3T + 3.41e6T^{2} \)
47 \( 1 + 1.57e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.90e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.13e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.44e3T + 1.38e7T^{2} \)
67 \( 1 - 3.49e3T + 2.01e7T^{2} \)
71 \( 1 - 1.93e3iT - 2.54e7T^{2} \)
73 \( 1 + 9.02e3T + 2.83e7T^{2} \)
79 \( 1 - 5.27e3T + 3.89e7T^{2} \)
83 \( 1 - 6.14e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 + 2.80e3T + 8.85e7T^{2} \)
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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

−16.14233121629841574068007163944, −13.99026056634987009935016140466, −13.24534142734715698563260044378, −11.81598166515821482872491228168, −11.07566588439315144152296955885, −9.551644204994504609191490054834, −8.429041909226720574549807435556, −5.12930890024279187535300812097, −3.30149216670597581976194195172, −1.05118465924749545793646867202, 4.52684156103309417973165359126, 6.24959575414367357224578006561, 7.40831841059815501859738818531, 8.708844452389173878328741915741, 10.33359541041103543273828866575, 12.60414387217061506931505873307, 14.08087565435155013744956907380, 14.90261983208475283173188195780, 15.78597545736317246807215294290, 17.12432981919685863795086223002

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