Properties

Label 2-3e3-9.7-c5-0-0
Degree $2$
Conductor $27$
Sign $-0.707 - 0.706i$
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  + (−1.78 + 3.09i)2-s + (9.59 + 16.6i)4-s + (−4.05 − 7.02i)5-s + (−87.7 + 151. i)7-s − 183.·8-s + 29.0·10-s + (−206. + 358. i)11-s + (−61.8 − 107. i)13-s + (−313. − 543. i)14-s + (20.9 − 36.2i)16-s + 2.22e3·17-s + 891.·19-s + (77.7 − 134. i)20-s + (−740. − 1.28e3i)22-s + (−293. − 509. i)23-s + ⋯
L(s)  = 1  + (−0.316 + 0.548i)2-s + (0.299 + 0.519i)4-s + (−0.0725 − 0.125i)5-s + (−0.676 + 1.17i)7-s − 1.01·8-s + 0.0917·10-s + (−0.515 + 0.892i)11-s + (−0.101 − 0.175i)13-s + (−0.428 − 0.741i)14-s + (0.0204 − 0.0354i)16-s + 1.86·17-s + 0.566·19-s + (0.0434 − 0.0753i)20-s + (−0.326 − 0.564i)22-s + (−0.115 − 0.200i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.707 - 0.706i$
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: \((2,\ 27,\ (\ :5/2),\ -0.707 - 0.706i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.404614 + 0.978316i\)
\(L(\frac12)\) \(\approx\) \(0.404614 + 0.978316i\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.78 - 3.09i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (4.05 + 7.02i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (87.7 - 151. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (206. - 358. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (61.8 + 107. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 2.22e3T + 1.41e6T^{2} \)
19 \( 1 - 891.T + 2.47e6T^{2} \)
23 \( 1 + (293. + 509. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.46e3 - 4.27e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.83e3 - 4.90e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.20e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.19e3 - 2.06e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-2.88e3 + 4.99e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-3.53e3 + 6.11e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.15e4T + 4.18e8T^{2} \)
59 \( 1 + (-4.87e3 - 8.44e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-932. + 1.61e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.96e4 - 3.40e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.36e4T + 1.80e9T^{2} \)
73 \( 1 + 5.21e4T + 2.07e9T^{2} \)
79 \( 1 + (2.53e4 - 4.39e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.78e4 - 4.81e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 4.65e4T + 5.58e9T^{2} \)
97 \( 1 + (-2.16e4 + 3.74e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−16.52910219858638152601224086522, −15.77060780198197113103743665299, −14.68416605844868058957937912313, −12.60767889573284679118622978845, −12.07627231790776220942665943469, −9.941846482130939201191931169104, −8.574083571874166040132126640724, −7.23564356320867364012290499316, −5.62051145978438426209054696480, −2.92760961878183949269406824455, 0.78402861444979102849448153660, 3.27822343676835652591638377216, 5.85496232582327190674969900033, 7.54163405524525781814676116930, 9.611497120482222711428763779132, 10.50985852438410000497934679823, 11.69880523038708290373307758134, 13.32113341565690534647170214673, 14.52149188954662374763634518957, 15.98453547913112200517963308383

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