Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.920 - 0.390i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.592 + 3.36i)2-s + (−5.15 + 0.614i)3-s + (−3.42 − 1.24i)4-s + (−8.41 + 7.05i)5-s + (0.992 − 17.7i)6-s + (4.14 − 1.50i)7-s + (−7.42 + 12.8i)8-s + (26.2 − 6.34i)9-s + (−18.7 − 32.4i)10-s + (44.1 + 37.0i)11-s + (18.4 + 4.32i)12-s + (6.80 + 38.6i)13-s + (2.61 + 14.8i)14-s + (39.0 − 41.5i)15-s + (−61.1 − 51.3i)16-s + (−44.7 − 77.4i)17-s + ⋯
L(s)  = 1  + (−0.209 + 1.18i)2-s + (−0.992 + 0.118i)3-s + (−0.428 − 0.155i)4-s + (−0.752 + 0.631i)5-s + (0.0675 − 1.20i)6-s + (0.223 − 0.0813i)7-s + (−0.328 + 0.568i)8-s + (0.972 − 0.234i)9-s + (−0.592 − 1.02i)10-s + (1.21 + 1.01i)11-s + (0.443 + 0.104i)12-s + (0.145 + 0.823i)13-s + (0.0498 + 0.282i)14-s + (0.672 − 0.715i)15-s + (−0.956 − 0.802i)16-s + (−0.638 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.920 - 0.390i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.920 - 0.390i)$
$L(2)$  $\approx$  $0.144224 + 0.710289i$
$L(\frac12)$  $\approx$  $0.144224 + 0.710289i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (5.15 - 0.614i)T \)
good2 \( 1 + (0.592 - 3.36i)T + (-7.51 - 2.73i)T^{2} \)
5 \( 1 + (8.41 - 7.05i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-4.14 + 1.50i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (-44.1 - 37.0i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (-6.80 - 38.6i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (44.7 + 77.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-29.6 + 51.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-21.4 - 7.81i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (10.2 - 58.0i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-313. - 114. i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (47.4 + 82.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (62.8 + 356. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-361. - 303. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (222. - 81.0i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 - 391.T + 1.48e5T^{2} \)
59 \( 1 + (-429. + 360. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (158. - 57.7i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (94.5 + 536. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (97.0 + 168. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (80.6 - 139. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-63.1 + 358. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (98.1 - 556. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (-187. + 324. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (446. + 374. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.30503945324959783049729899976, −16.09196439505006584084233837628, −15.34926261309240068898435120914, −14.18071235871343662939649572348, −11.95155998677233385220224852843, −11.24835414794378669312802790621, −9.283191562313091629147197717449, −7.25239608185441677333853431128, −6.64190141612858334722241827442, −4.64736029416186619764215571201, 0.941049926643177377366575243940, 4.03325196703072359429278301910, 6.21388934444744618113793531438, 8.431376420183442030142998978930, 10.19780165562194632737292945885, 11.42682065278233527575734097557, 12.00995356787666590487155397342, 13.17532280454961416937206895749, 15.36240245711233726513369485056, 16.51090354761320894810358685319

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