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Properties

Label 2-3e3-9.7-c5-0-2
Degree $2$
Conductor $27$
Sign $0.624 + 0.781i$
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  + (−4.96 + 8.59i)2-s + (−33.2 − 57.5i)4-s + (−23.6 − 40.9i)5-s + (1.01 − 1.75i)7-s + 342.·8-s + 469.·10-s + (91.8 − 159. i)11-s + (−364. − 631. i)13-s + (10.0 + 17.4i)14-s + (−633. + 1.09e3i)16-s − 1.21e3·17-s − 473.·19-s + (−1.57e3 + 2.72e3i)20-s + (911. + 1.57e3i)22-s + (−1.80e3 − 3.12e3i)23-s + ⋯
L(s)  = 1  + (−0.877 + 1.51i)2-s + (−1.03 − 1.79i)4-s + (−0.423 − 0.732i)5-s + (0.00783 − 0.0135i)7-s + 1.88·8-s + 1.48·10-s + (0.228 − 0.396i)11-s + (−0.598 − 1.03i)13-s + (0.0137 + 0.0237i)14-s + (−0.618 + 1.07i)16-s − 1.01·17-s − 0.300·19-s + (−0.879 + 1.52i)20-s + (0.401 + 0.695i)22-s + (−0.712 − 1.23i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(4.33036\)
Root analytic conductor: \(2.08095\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :5/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.363706 - 0.174987i\)
\(L(\frac12)\) \(\approx\) \(0.363706 - 0.174987i\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.96 - 8.59i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (23.6 + 40.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-1.01 + 1.75i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-91.8 + 159. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (364. + 631. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.21e3T + 1.41e6T^{2} \)
19 \( 1 + 473.T + 2.47e6T^{2} \)
23 \( 1 + (1.80e3 + 3.12e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (663. - 1.14e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.59e3 - 4.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.47e4T + 6.93e7T^{2} \)
41 \( 1 + (-3.15e3 - 5.47e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (3.06e3 - 5.31e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-1.58e3 + 2.74e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.22e4T + 4.18e8T^{2} \)
59 \( 1 + (1.47e4 + 2.55e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.01e4 + 3.49e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.17e4 + 2.03e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 123.T + 1.80e9T^{2} \)
73 \( 1 - 3.52e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.40e4 + 4.16e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-5.16e3 + 8.95e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 4.25e4T + 5.58e9T^{2} \)
97 \( 1 + (4.93e4 - 8.54e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−16.17556227891436886772278277477, −15.35000260611722079153078641819, −14.12242269953910557021813574309, −12.48385949071762321368969482914, −10.42355697115424458867672928585, −8.883612146005484857639628015202, −8.045354040053222393593315856916, −6.52417911563383234671153133633, −4.91131768982226875487182047134, −0.34664886680508589895978091700, 2.13547312892609978783312274640, 3.93830320767800756980101993002, 7.21368839953274196923949498578, 8.898426971627851601277822230513, 10.08726740965907411020434006875, 11.30379106880693496905354963603, 12.07949633948394923920679913519, 13.62861540678454407002252586752, 15.25953528070165264586546297234, 17.02966596147454400782077090902

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