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Properties

Label 2-3e3-9.5-c6-0-0
Degree $2$
Conductor $27$
Sign $-0.800 - 0.598i$
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  + (−10.6 + 6.14i)2-s + (43.5 − 75.4i)4-s + (157. + 90.9i)5-s + (83.3 + 144. i)7-s + 284. i·8-s − 2.23e3·10-s + (−1.48e3 + 859. i)11-s + (−279. + 483. i)13-s + (−1.77e3 − 1.02e3i)14-s + (1.04e3 + 1.80e3i)16-s + 5.31e3i·17-s − 1.40e3·19-s + (1.37e4 − 7.92e3i)20-s + (1.05e4 − 1.82e4i)22-s + (−3.65e3 − 2.11e3i)23-s + ⋯
L(s)  = 1  + (−1.33 + 0.768i)2-s + (0.680 − 1.17i)4-s + (1.26 + 0.727i)5-s + (0.243 + 0.420i)7-s + 0.554i·8-s − 2.23·10-s + (−1.11 + 0.645i)11-s + (−0.127 + 0.220i)13-s + (−0.646 − 0.373i)14-s + (0.254 + 0.440i)16-s + 1.08i·17-s − 0.205·19-s + (1.71 − 0.990i)20-s + (0.992 − 1.71i)22-s + (−0.300 − 0.173i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.800 - 0.598i$
Analytic conductor: \(6.21146\)
Root analytic conductor: \(2.49228\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3),\ -0.800 - 0.598i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.253093 + 0.760978i\)
\(L(\frac12)\) \(\approx\) \(0.253093 + 0.760978i\)
\(L(4)\) 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 \( 1 + (10.6 - 6.14i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (-157. - 90.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-83.3 - 144. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (1.48e3 - 859. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (279. - 483. i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 - 5.31e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.40e3T + 4.70e7T^{2} \)
23 \( 1 + (3.65e3 + 2.11e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.79e3 + 1.61e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.22e4 - 3.85e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 6.45e3T + 2.56e9T^{2} \)
41 \( 1 + (-5.32e4 - 3.07e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (1.13e4 + 1.97e4i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-3.89e4 + 2.24e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + 7.23e4iT - 2.21e10T^{2} \)
59 \( 1 + (-1.75e5 - 1.01e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.81e5 - 3.13e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-2.44e5 + 4.23e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 1.52e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.78e4T + 1.51e11T^{2} \)
79 \( 1 + (-1.46e5 - 2.52e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (3.56e5 - 2.05e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 7.58e5iT - 4.96e11T^{2} \)
97 \( 1 + (4.77e5 + 8.27e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−16.81869917211497573724579881699, −15.47095322939679282729069404067, −14.47298457066503607157948659340, −12.87237866277921174475128554812, −10.61893661589677240911167411757, −9.872841825976361624097412582948, −8.496318199687821929336286292047, −7.02699803125012447633325507178, −5.73271631603798948702771871907, −2.03636367378210291401885730095, 0.69191668625557917223196727374, 2.34891092746789853908352260907, 5.42590193696201246188202379446, 7.80962678864054563984616622547, 9.136193141128396622571573689716, 10.08431963501909820562395602738, 11.20618975006303804711864073050, 12.86757797049974610629973603916, 13.98704424139306314592605944649, 16.15679215389925604321266646263

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