Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.0369 - 0.999i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.932 + 0.339i)2-s + (1.23 + 5.04i)3-s + (−5.37 + 4.50i)4-s + (0.0250 + 0.142i)5-s + (−2.86 − 4.28i)6-s + (19.4 + 16.3i)7-s + (7.44 − 12.8i)8-s + (−23.9 + 12.4i)9-s + (−0.0716 − 0.124i)10-s + (11.3 − 64.1i)11-s + (−29.3 − 21.5i)12-s + (24.9 + 9.07i)13-s + (−23.6 − 8.60i)14-s + (−0.687 + 0.302i)15-s + (7.18 − 40.7i)16-s + (34.5 + 59.7i)17-s + ⋯
L(s)  = 1  + (−0.329 + 0.119i)2-s + (0.237 + 0.971i)3-s + (−0.671 + 0.563i)4-s + (0.00224 + 0.0127i)5-s + (−0.194 − 0.291i)6-s + (1.04 + 0.880i)7-s + (0.329 − 0.570i)8-s + (−0.887 + 0.461i)9-s + (−0.00226 − 0.00392i)10-s + (0.309 − 1.75i)11-s + (−0.707 − 0.518i)12-s + (0.531 + 0.193i)13-s + (−0.451 − 0.164i)14-s + (−0.0118 + 0.00520i)15-s + (0.112 − 0.636i)16-s + (0.492 + 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0369 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.0369 - 0.999i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.0369 - 0.999i)$
$L(2)$  $\approx$  $0.719564 + 0.693446i$
$L(\frac12)$  $\approx$  $0.719564 + 0.693446i$
$L(\frac{5}{2})$   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 \( 1 + (-1.23 - 5.04i)T \)
good2 \( 1 + (0.932 - 0.339i)T + (6.12 - 5.14i)T^{2} \)
5 \( 1 + (-0.0250 - 0.142i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (-19.4 - 16.3i)T + (59.5 + 337. i)T^{2} \)
11 \( 1 + (-11.3 + 64.1i)T + (-1.25e3 - 455. i)T^{2} \)
13 \( 1 + (-24.9 - 9.07i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (-34.5 - 59.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (15.9 - 27.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (25.9 - 21.7i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (-40.6 + 14.7i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (48.7 - 40.9i)T + (5.17e3 - 2.93e4i)T^{2} \)
37 \( 1 + (128. + 223. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (356. + 129. i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (-51.2 + 290. i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (228. + 192. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 - 340.T + 1.48e5T^{2} \)
59 \( 1 + (-53.1 - 301. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-515. - 432. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-161. - 58.9i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (523. + 906. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-87.6 + 151. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (85.3 - 31.0i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (909. - 331. i)T + (4.38e5 - 3.67e5i)T^{2} \)
89 \( 1 + (240. - 415. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (174. - 989. i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−16.98345138794045873077040704184, −16.13688808656964874565728701102, −14.70379045273005574427794842351, −13.72387818284634534560228055526, −11.88404327741466389458582955181, −10.57713736018898831811617844096, −8.724014336389074406853627664069, −8.444435427866871967735336446020, −5.48618464077367132410809413578, −3.68404877864825242955949749368, 1.39200601833495428808173821772, 4.81175170423804531728655267660, 7.06728266574116615067446174910, 8.359956587248931454735554303639, 9.896618263852847452989231614526, 11.39563547534543422081885299939, 12.94041937821128327960708515437, 14.11414746299064696156907905342, 14.85657400305062862431305484705, 17.17683556067010556205965552568

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