Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.969 - 0.244i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.82 + 0.663i)2-s + (5.12 + 0.878i)3-s + (−3.24 − 2.72i)4-s + (−0.470 + 2.66i)5-s + (8.75 + 5.00i)6-s + (−9.12 + 7.66i)7-s + (−11.8 − 20.5i)8-s + (25.4 + 8.99i)9-s + (−2.62 + 4.55i)10-s + (−3.88 − 22.0i)11-s + (−14.2 − 16.7i)12-s + (−63.1 + 22.9i)13-s + (−21.7 + 7.90i)14-s + (−4.75 + 13.2i)15-s + (−2.11 − 12.0i)16-s + (44.8 − 77.6i)17-s + ⋯
L(s)  = 1  + (0.644 + 0.234i)2-s + (0.985 + 0.169i)3-s + (−0.405 − 0.340i)4-s + (−0.0420 + 0.238i)5-s + (0.595 + 0.340i)6-s + (−0.492 + 0.413i)7-s + (−0.524 − 0.908i)8-s + (0.942 + 0.333i)9-s + (−0.0831 + 0.143i)10-s + (−0.106 − 0.604i)11-s + (−0.342 − 0.403i)12-s + (−1.34 + 0.490i)13-s + (−0.414 + 0.150i)14-s + (−0.0818 + 0.228i)15-s + (−0.0330 − 0.187i)16-s + (0.639 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.969 - 0.244i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.969 - 0.244i)$
$L(2)$  $\approx$  $1.70672 + 0.211636i$
$L(\frac12)$  $\approx$  $1.70672 + 0.211636i$
$L(\frac{5}{2})$   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 \( 1 + (-5.12 - 0.878i)T \)
good2 \( 1 + (-1.82 - 0.663i)T + (6.12 + 5.14i)T^{2} \)
5 \( 1 + (0.470 - 2.66i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (9.12 - 7.66i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (3.88 + 22.0i)T + (-1.25e3 + 455. i)T^{2} \)
13 \( 1 + (63.1 - 22.9i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (-44.8 + 77.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-39.1 - 67.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76.3 - 64.0i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (-102. - 37.2i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (170. + 143. i)T + (5.17e3 + 2.93e4i)T^{2} \)
37 \( 1 + (38.3 - 66.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (357. - 129. i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (80.7 + 458. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (-289. + 242. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + 0.995T + 1.48e5T^{2} \)
59 \( 1 + (-111. + 632. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (282. - 237. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (401. - 146. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-29.7 + 51.5i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (157. + 272. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (164. + 59.8i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (764. + 278. i)T + (4.38e5 + 3.67e5i)T^{2} \)
89 \( 1 + (-511. - 885. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-140. - 794. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−16.50730502934276125759488466456, −15.28273710681809341460128486754, −14.40508581413683879740223984071, −13.53460251307070780283253970362, −12.22682905836856033488255042627, −10.00534586777917014450784571960, −9.070825158686441749123784835226, −7.16221990006997904664721317489, −5.16558197868848163474676995912, −3.25061966716389201388190099136, 3.02296171969181534676500966324, 4.69808224630632913066629481721, 7.31170991489639171370365139433, 8.732981774520848616449178179915, 10.10486102174350433030335438297, 12.42254269808233929489054793924, 12.93000621882889094833766995947, 14.24085591366675041129028298546, 15.11504472738345042636088748361, 16.83948002733588413165276034571

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