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Properties

Label 2-3e3-9.4-c5-0-2
Degree $2$
Conductor $27$
Sign $0.993 + 0.114i$
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  + (1.47 + 2.55i)2-s + (11.6 − 20.1i)4-s + (32.4 − 56.1i)5-s + (80.0 + 138. i)7-s + 163.·8-s + 191.·10-s + (−103. − 178. i)11-s + (32.8 − 56.9i)13-s + (−236. + 409. i)14-s + (−131. − 227. i)16-s + 601.·17-s − 2.00e3·19-s + (−754. − 1.30e3i)20-s + (304. − 528. i)22-s + (−2.18e3 + 3.78e3i)23-s + ⋯
L(s)  = 1  + (0.261 + 0.452i)2-s + (0.363 − 0.629i)4-s + (0.580 − 1.00i)5-s + (0.617 + 1.07i)7-s + 0.901·8-s + 0.605·10-s + (−0.257 − 0.445i)11-s + (0.0539 − 0.0934i)13-s + (−0.322 + 0.558i)14-s + (−0.128 − 0.222i)16-s + 0.504·17-s − 1.27·19-s + (−0.421 − 0.730i)20-s + (0.134 − 0.232i)22-s + (−0.861 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.993 + 0.114i$
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: \((2,\ 27,\ (\ :5/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.03231 - 0.116974i\)
\(L(\frac12)\) \(\approx\) \(2.03231 - 0.116974i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1.47 - 2.55i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-32.4 + 56.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-80.0 - 138. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (103. + 178. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-32.8 + 56.9i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 601.T + 1.41e6T^{2} \)
19 \( 1 + 2.00e3T + 2.47e6T^{2} \)
23 \( 1 + (2.18e3 - 3.78e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (467. + 809. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.62e3 - 4.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 5.68e3T + 6.93e7T^{2} \)
41 \( 1 + (3.22e3 - 5.58e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.40e3 - 2.43e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-3.90e3 - 6.76e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 3.22e4T + 4.18e8T^{2} \)
59 \( 1 + (-7.84e3 + 1.35e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.11e4 + 1.93e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-9.45e3 + 1.63e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.67e4T + 1.80e9T^{2} \)
73 \( 1 - 4.53e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.56e4 - 2.70e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.09e4 + 5.35e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 1.21e4T + 5.58e9T^{2} \)
97 \( 1 + (3.67e4 + 6.36e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−16.11511141254610028379014701594, −15.11750075819191628805799683920, −13.94618357193520336457852341225, −12.63304754041091707559956002360, −11.16499080387105510116208328396, −9.538878578546512081613044876212, −8.144868168603859136004584506927, −5.98908495764928910896893669677, −5.07768881095233666089705061209, −1.72297001227169356495419369837, 2.27037253714779250498393950444, 4.15443477876413838454745383529, 6.67398525535259677539452475009, 7.923380027866324995016681955924, 10.30521379601396833998772235960, 10.98598244161835331364415768637, 12.53386323626020230988443972224, 13.80762469178097728437153218149, 14.78150760971475461250476571801, 16.55898905040198178595031938177

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