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  :  $\chi_{27} (4, \cdot )$
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

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

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