Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.987 + 0.157i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.06 − 1.48i)2-s + (−1.65 + 4.92i)3-s + (8.22 − 6.90i)4-s + (−0.745 − 4.22i)5-s + (0.561 + 22.4i)6-s + (−15.6 − 13.1i)7-s + (5.92 − 10.2i)8-s + (−21.5 − 16.3i)9-s + (−9.29 − 16.0i)10-s + (−4.83 + 27.4i)11-s + (20.3 + 51.9i)12-s + (84.9 + 30.9i)13-s + (−83.1 − 30.2i)14-s + (22.0 + 3.32i)15-s + (−6.01 + 34.0i)16-s + (−37.8 − 65.5i)17-s + ⋯
L(s)  = 1  + (1.43 − 0.523i)2-s + (−0.318 + 0.947i)3-s + (1.02 − 0.862i)4-s + (−0.0666 − 0.378i)5-s + (0.0382 + 1.52i)6-s + (−0.845 − 0.709i)7-s + (0.261 − 0.453i)8-s + (−0.797 − 0.603i)9-s + (−0.293 − 0.509i)10-s + (−0.132 + 0.751i)11-s + (0.490 + 1.24i)12-s + (1.81 + 0.659i)13-s + (−1.58 − 0.577i)14-s + (0.379 + 0.0572i)15-s + (−0.0939 + 0.532i)16-s + (−0.540 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.987 + 0.157i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.987 + 0.157i)$
$L(2)$  $\approx$  $1.93629 - 0.153087i$
$L(\frac12)$  $\approx$  $1.93629 - 0.153087i$
$L(\frac{5}{2})$   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 \( 1 + (1.65 - 4.92i)T \)
good2 \( 1 + (-4.06 + 1.48i)T + (6.12 - 5.14i)T^{2} \)
5 \( 1 + (0.745 + 4.22i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (15.6 + 13.1i)T + (59.5 + 337. i)T^{2} \)
11 \( 1 + (4.83 - 27.4i)T + (-1.25e3 - 455. i)T^{2} \)
13 \( 1 + (-84.9 - 30.9i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (37.8 + 65.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-32.9 + 57.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (109. - 92.0i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (-17.6 + 6.44i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (26.2 - 22.0i)T + (5.17e3 - 2.93e4i)T^{2} \)
37 \( 1 + (62.6 + 108. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-54.4 - 19.8i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (16.8 - 95.3i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (80.7 + 67.7i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + 603.T + 1.48e5T^{2} \)
59 \( 1 + (-72.3 - 410. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (413. + 347. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-57.3 - 20.8i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (50.1 + 86.9i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-277. + 480. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-297. + 108. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (-363. + 132. i)T + (4.38e5 - 3.67e5i)T^{2} \)
89 \( 1 + (-566. + 981. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (194. - 1.10e3i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−16.22938460067001072059590490572, −15.63757285720007028918391240301, −14.06011297794350741471984812055, −13.18753217928031990522485687343, −11.79247598761245451035821730689, −10.74864773935917015206438576746, −9.229538257169527888161794961282, −6.37171089038439890381696102700, −4.72952413724209369512835963878, −3.55573961102633337467375110131, 3.28647041883505770858168963711, 5.86437896016960237304328026674, 6.40820676823645426923043733612, 8.324419750123768993548844907592, 10.96954601810979569879803669405, 12.40326244957679552362767494485, 13.16248771326826877159435817410, 14.14368731719241256674261369721, 15.55122230096995345987622988627, 16.41535251156091718936632666892

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