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Properties

Label 2-3e3-9.5-c4-0-0
Degree $2$
Conductor $27$
Sign $0.463 - 0.886i$
Analytic cond. $2.79098$
Root an. cond. $1.67062$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.307 + 0.177i)2-s + (−7.93 + 13.7i)4-s + (30.0 + 17.3i)5-s + (15.6 + 27.0i)7-s − 11.3i·8-s − 12.3·10-s + (−49.9 + 28.8i)11-s + (36.6 − 63.4i)13-s + (−9.60 − 5.54i)14-s + (−124. − 216. i)16-s − 386. i·17-s + 115.·19-s + (−477. + 275. i)20-s + (10.2 − 17.7i)22-s + (474. + 274. i)23-s + ⋯
L(s)  = 1  + (−0.0769 + 0.0443i)2-s + (−0.496 + 0.859i)4-s + (1.20 + 0.694i)5-s + (0.318 + 0.551i)7-s − 0.176i·8-s − 0.123·10-s + (−0.413 + 0.238i)11-s + (0.216 − 0.375i)13-s + (−0.0489 − 0.0282i)14-s + (−0.488 − 0.845i)16-s − 1.33i·17-s + 0.320·19-s + (−1.19 + 0.689i)20-s + (0.0211 − 0.0366i)22-s + (0.897 + 0.518i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.463 - 0.886i$
Analytic conductor: \(2.79098\)
Root analytic conductor: \(1.67062\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :2),\ 0.463 - 0.886i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.17045 + 0.708665i\)
\(L(\frac12)\) \(\approx\) \(1.17045 + 0.708665i\)
\(L(3)\) 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 \( 1 + (0.307 - 0.177i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + (-30.0 - 17.3i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-15.6 - 27.0i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (49.9 - 28.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-36.6 + 63.4i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 386. iT - 8.35e4T^{2} \)
19 \( 1 - 115.T + 1.30e5T^{2} \)
23 \( 1 + (-474. - 274. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-680. + 392. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (272. - 471. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 898.T + 1.87e6T^{2} \)
41 \( 1 + (2.24e3 + 1.29e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.00e3 + 1.73e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-702. + 405. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.22e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.30e3 + 756. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-951. - 1.64e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.25e3 - 3.90e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 3.99e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.43e3T + 2.83e7T^{2} \)
79 \( 1 + (601. + 1.04e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (8.01e3 - 4.62e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.92e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.33e3 + 5.77e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−17.08956399497630551455476358477, −15.57973663801880420622320709404, −14.07451368861290281645479365398, −13.22817235342129987792120710858, −11.76561951407405739449157081825, −10.08938202497819015722906171952, −8.843897739633960053311887313114, −7.17363978033986027122903695454, −5.28773074098139671210482401637, −2.78128447734291669731027954066, 1.39120320666222587972302758615, 4.81829482549395935130625318206, 6.15829478421008039014518690371, 8.556989562239951931605268903464, 9.786064846712154736107540745221, 10.84041162441304333400938403796, 12.97375083851961892284065594836, 13.78173937668406255091850004717, 14.89632133210085305553881447912, 16.62737742190966846557517514346

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