Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.849 - 0.528i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.111 − 0.634i)2-s + (−0.446 + 5.17i)3-s + (7.12 + 2.59i)4-s + (−0.393 + 0.330i)5-s + (3.23 + 0.862i)6-s + (9.02 − 3.28i)7-s + (5.02 − 8.69i)8-s + (−26.6 − 4.62i)9-s + (0.165 + 0.287i)10-s + (−31.1 − 26.1i)11-s + (−16.6 + 35.7i)12-s + (−0.304 − 1.72i)13-s + (−1.07 − 6.09i)14-s + (−1.53 − 2.18i)15-s + (41.5 + 34.8i)16-s + (−37.5 − 65.1i)17-s + ⋯
L(s)  = 1  + (0.0395 − 0.224i)2-s + (−0.0859 + 0.996i)3-s + (0.890 + 0.324i)4-s + (−0.0352 + 0.0295i)5-s + (0.220 + 0.0587i)6-s + (0.487 − 0.177i)7-s + (0.221 − 0.384i)8-s + (−0.985 − 0.171i)9-s + (0.00524 + 0.00907i)10-s + (−0.853 − 0.716i)11-s + (−0.399 + 0.859i)12-s + (−0.00649 − 0.0368i)13-s + (−0.0205 − 0.116i)14-s + (−0.0264 − 0.0376i)15-s + (0.648 + 0.544i)16-s + (−0.536 − 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.849 - 0.528i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.849 - 0.528i)$
$L(2)$  $\approx$  $1.27876 + 0.365202i$
$L(\frac12)$  $\approx$  $1.27876 + 0.365202i$
$L(\frac{5}{2})$   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 \( 1 + (0.446 - 5.17i)T \)
good2 \( 1 + (-0.111 + 0.634i)T + (-7.51 - 2.73i)T^{2} \)
5 \( 1 + (0.393 - 0.330i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-9.02 + 3.28i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (31.1 + 26.1i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (0.304 + 1.72i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (37.5 + 65.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-42.1 + 73.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-121. - 44.1i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (45.3 - 257. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (284. + 103. i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (103. + 178. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-54.5 - 309. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-99.6 - 83.6i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (186. - 67.9i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 - 527.T + 1.48e5T^{2} \)
59 \( 1 + (59.2 - 49.6i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (-251. + 91.6i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (-15.6 - 88.6i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (360. + 624. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-66.7 + 115. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-21.9 + 124. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-146. + 832. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (693. - 1.20e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (216. + 181. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−16.62924314926765729276916858198, −15.86664618944562335112541757994, −14.79435890242326944299879885638, −13.17013554700411165874453723104, −11.30150583099460447880142734439, −10.92132347982990976340358438547, −9.135113145889426411485655586986, −7.37610840207605996895759657387, −5.25658111834266303438356147008, −3.14504001774021647626870029415, 2.07480092707585363972836243542, 5.57477092573657061416331953334, 7.04195511122095828166989857045, 8.205429935617050613071804736453, 10.49992894263189078625673619559, 11.74400483949329772795042915685, 12.87677340365525254838663578896, 14.39045049665587174673391432890, 15.41468877053338737685903275087, 16.82967379590989156091322296531

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