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Properties

Label 2-3e3-9.4-c5-0-1
Degree $2$
Conductor $27$
Sign $-0.646 - 0.763i$
Analytic cond. $4.33036$
Root an. cond. $2.08095$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.77 + 6.53i)2-s + (−12.4 + 21.6i)4-s + (−43.7 + 75.7i)5-s + (20.6 + 35.6i)7-s + 52.9·8-s − 660.·10-s + (−3.84 − 6.65i)11-s + (302. − 524. i)13-s + (−155. + 269. i)14-s + (599. + 1.03e3i)16-s + 566.·17-s + 2.06e3·19-s + (−1.09e3 − 1.89e3i)20-s + (29.0 − 50.2i)22-s + (−137. + 237. i)23-s + ⋯
L(s)  = 1  + (0.667 + 1.15i)2-s + (−0.390 + 0.676i)4-s + (−0.782 + 1.35i)5-s + (0.158 + 0.275i)7-s + 0.292·8-s − 2.08·10-s + (−0.00957 − 0.0165i)11-s + (0.496 − 0.860i)13-s + (−0.212 + 0.367i)14-s + (0.585 + 1.01i)16-s + 0.475·17-s + 1.30·19-s + (−0.610 − 1.05i)20-s + (0.0127 − 0.0221i)22-s + (−0.0540 + 0.0936i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(4.33036\)
Root analytic conductor: \(2.08095\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :5/2),\ -0.646 - 0.763i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.804488 + 1.73563i\)
\(L(\frac12)\) \(\approx\) \(0.804488 + 1.73563i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.77 - 6.53i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (43.7 - 75.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-20.6 - 35.6i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (3.84 + 6.65i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-302. + 524. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 566.T + 1.41e6T^{2} \)
19 \( 1 - 2.06e3T + 2.47e6T^{2} \)
23 \( 1 + (137. - 237. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.41e3 + 4.17e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.23e3 - 3.86e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 242.T + 6.93e7T^{2} \)
41 \( 1 + (1.75e3 - 3.03e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (4.26e3 + 7.39e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (8.99e3 + 1.55e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 2.02e3T + 4.18e8T^{2} \)
59 \( 1 + (-1.00e3 + 1.74e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.41e4 - 2.45e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.09e4 - 3.62e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.26e4T + 1.80e9T^{2} \)
73 \( 1 - 3.28e4T + 2.07e9T^{2} \)
79 \( 1 + (4.40e4 + 7.63e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (5.28e3 + 9.16e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 1.24e5T + 5.58e9T^{2} \)
97 \( 1 + (-8.14e4 - 1.41e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−16.23282909748334064548623840417, −15.32017478594499220877522654791, −14.63327261277156932524415115584, −13.47778890224467386520419621997, −11.70071649164421833673533097056, −10.39567894570926747383750752029, −7.993586520029643741456425490544, −7.00555137673337028991636557574, −5.55775571991371036061199183495, −3.50040640557136259246343045863, 1.23878997270571428976326488877, 3.74983232726867188289746897270, 4.98988539499800165802634938327, 7.76492192232203655311200292942, 9.379670987117012945741785038802, 11.16322324603984528552894810434, 12.04778369353807049590808491586, 12.99939389558786319848228955412, 14.15133077233505589198385998738, 16.01745037198158229887124768312

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