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Properties

Label 2-3e3-9.7-c3-0-0
Degree $2$
Conductor $27$
Sign $0.450 - 0.892i$
Analytic cond. $1.59305$
Root an. cond. $1.26216$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 + 1.18i)2-s + (3.05 + 5.29i)4-s + (5.18 + 8.98i)5-s + (2.55 − 4.43i)7-s − 19.3·8-s − 14.2·10-s + (27.9 − 48.4i)11-s + (−18.7 − 32.5i)13-s + (3.51 + 6.08i)14-s + (−11.1 + 19.3i)16-s − 23.6·17-s + 39.0·19-s + (−31.7 + 54.9i)20-s + (38.4 + 66.5i)22-s + (35.5 + 61.5i)23-s + ⋯
L(s)  = 1  + (−0.242 + 0.420i)2-s + (0.382 + 0.662i)4-s + (0.463 + 0.803i)5-s + (0.138 − 0.239i)7-s − 0.856·8-s − 0.450·10-s + (0.767 − 1.32i)11-s + (−0.400 − 0.694i)13-s + (0.0670 + 0.116i)14-s + (−0.174 + 0.302i)16-s − 0.337·17-s + 0.471·19-s + (−0.354 + 0.614i)20-s + (0.372 + 0.644i)22-s + (0.322 + 0.558i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(1.59305\)
Root analytic conductor: \(1.26216\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.997923 + 0.614182i\)
\(L(\frac12)\) \(\approx\) \(0.997923 + 0.614182i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.686 - 1.18i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-5.18 - 8.98i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-2.55 + 4.43i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-27.9 + 48.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.7 + 32.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 23.6T + 4.91e3T^{2} \)
19 \( 1 - 39.0T + 6.85e3T^{2} \)
23 \( 1 + (-35.5 - 61.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (14.1 - 24.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (6.44 + 11.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 180.T + 5.06e4T^{2} \)
41 \( 1 + (107. + 186. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (30.6 - 53.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (30.9 - 53.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 492.T + 1.48e5T^{2} \)
59 \( 1 + (-394. - 683. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (260. - 451. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (152. + 263. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 270.T + 3.57e5T^{2} \)
73 \( 1 + 925.T + 3.89e5T^{2} \)
79 \( 1 + (-644. + 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-356. + 618. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 404.T + 7.04e5T^{2} \)
97 \( 1 + (37.5 - 64.9i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−17.11833025183647360129215583401, −15.99128080841003869115235296486, −14.69726443765428561638949071584, −13.50583292796908190710228440815, −11.86327302563244465854297191810, −10.66232324990546059399081861357, −8.890230962037475437235582083283, −7.39403740343528398650769457240, −6.09338521980883979071422840977, −3.17977023801159720974437789391, 1.78104682209893045679366147612, 4.97360334988533263775153055276, 6.75591990290597958989807088574, 9.047807265880581191928704155680, 9.909135239008605228644967010698, 11.55350226870938991350607049215, 12.59775759647915255950700959927, 14.28002952024467194099102464533, 15.34217883305334295720747373215, 16.78225107047225004913406058615

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