Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.320 - 0.947i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 2.04i)2-s + (−2.73 + 4.41i)3-s + (0.375 + 2.12i)4-s + (6.73 + 2.45i)5-s + (−15.7 + 5.17i)6-s + (0.843 − 4.78i)7-s + (9.30 − 16.1i)8-s + (−12.0 − 24.1i)9-s + (11.4 + 19.7i)10-s + (20.1 − 7.34i)11-s + (−10.4 − 4.16i)12-s + (−26.7 + 22.4i)13-s + (11.8 − 9.95i)14-s + (−29.2 + 23.0i)15-s + (71.9 − 26.2i)16-s + (−57.1 − 99.0i)17-s + ⋯
L(s)  = 1  + (0.863 + 0.724i)2-s + (−0.526 + 0.849i)3-s + (0.0469 + 0.266i)4-s + (0.602 + 0.219i)5-s + (−1.07 + 0.352i)6-s + (0.0455 − 0.258i)7-s + (0.411 − 0.712i)8-s + (−0.444 − 0.895i)9-s + (0.361 + 0.625i)10-s + (0.553 − 0.201i)11-s + (−0.250 − 0.100i)12-s + (−0.570 + 0.478i)13-s + (0.226 − 0.189i)14-s + (−0.503 + 0.396i)15-s + (1.12 − 0.409i)16-s + (−0.815 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.320 - 0.947i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (16, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.320 - 0.947i)$
$L(2)$  $\approx$  $1.28367 + 0.920568i$
$L(\frac12)$  $\approx$  $1.28367 + 0.920568i$
$L(\frac{5}{2})$   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 \( 1 + (2.73 - 4.41i)T \)
good2 \( 1 + (-2.44 - 2.04i)T + (1.38 + 7.87i)T^{2} \)
5 \( 1 + (-6.73 - 2.45i)T + (95.7 + 80.3i)T^{2} \)
7 \( 1 + (-0.843 + 4.78i)T + (-322. - 117. i)T^{2} \)
11 \( 1 + (-20.1 + 7.34i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (26.7 - 22.4i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (57.1 + 99.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (75.8 - 131. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.1 - 91.4i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-144. - 121. i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (1.10 + 6.29i)T + (-2.79e4 + 1.01e4i)T^{2} \)
37 \( 1 + (196. + 340. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-159. + 134. i)T + (1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (142. - 51.9i)T + (6.09e4 - 5.11e4i)T^{2} \)
47 \( 1 + (29.2 - 165. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 2.81T + 1.48e5T^{2} \)
59 \( 1 + (-231. - 84.1i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (123. - 702. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-181. + 152. i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (61.2 + 106. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (137. - 238. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-197. - 166. i)T + (8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (-1.01e3 - 855. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (41.5 - 72.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-34.8 + 12.6i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−16.70238838839537369405565232331, −15.84249364163678431938654153621, −14.54945856616171914915631493625, −13.88579575313490249107121612562, −12.15530951953707669071916275321, −10.55424390650720007932453770225, −9.390485940742395624203294413474, −6.83075424556446502633622633310, −5.59974459069828088722651486801, −4.17234169222354352345133373495, 2.20208299318965194795403470130, 4.81496866556960304674423532184, 6.44671050287568610252670457485, 8.443022470730587200712291511970, 10.61840259167332728436997269453, 11.84665225878326324646365945410, 12.86274679758233798862561057509, 13.54319607961033676603123023863, 15.01048769650089866625665554054, 17.23684791760929743827253611519

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