Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.711 - 0.702i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.883 + 0.741i)2-s + (−5.10 + 0.962i)3-s + (−1.15 + 6.56i)4-s + (−7.21 + 2.62i)5-s + (3.79 − 4.63i)6-s + (2.25 + 12.7i)7-s + (−8.46 − 14.6i)8-s + (25.1 − 9.82i)9-s + (4.42 − 7.67i)10-s + (−10.9 − 3.99i)11-s + (−0.407 − 34.6i)12-s + (54.8 + 46.0i)13-s + (−11.4 − 9.61i)14-s + (34.3 − 20.3i)15-s + (−31.7 − 11.5i)16-s + (−18.1 + 31.4i)17-s + ⋯
L(s)  = 1  + (−0.312 + 0.262i)2-s + (−0.982 + 0.185i)3-s + (−0.144 + 0.820i)4-s + (−0.645 + 0.234i)5-s + (0.258 − 0.315i)6-s + (0.121 + 0.689i)7-s + (−0.373 − 0.647i)8-s + (0.931 − 0.364i)9-s + (0.140 − 0.242i)10-s + (−0.300 − 0.109i)11-s + (−0.00979 − 0.833i)12-s + (1.16 + 0.981i)13-s + (−0.218 − 0.183i)14-s + (0.590 − 0.350i)15-s + (−0.496 − 0.180i)16-s + (−0.258 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.711 - 0.702i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.711 - 0.702i)$
$L(2)$  $\approx$  $0.211738 + 0.515667i$
$L(\frac12)$  $\approx$  $0.211738 + 0.515667i$
$L(\frac{5}{2})$   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 \( 1 + (5.10 - 0.962i)T \)
good2 \( 1 + (0.883 - 0.741i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (7.21 - 2.62i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-2.25 - 12.7i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (10.9 + 3.99i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (-54.8 - 46.0i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (18.1 - 31.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (11.8 + 20.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.5 - 196. i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-218. + 183. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-5.40 + 30.6i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-171. + 296. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-188. - 158. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (143. + 52.3i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-49.8 - 282. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 321.T + 1.48e5T^{2} \)
59 \( 1 + (382. - 139. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-36.2 - 205. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (152. + 127. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (82.6 - 143. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-484. - 838. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-759. + 637. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-440. + 369. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (140. + 243. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-446. - 162. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−17.35038290433287415225842146211, −16.02009753312903360268546954330, −15.53588902079240456704400532047, −13.34469067144779348811851830116, −11.96947591199880727492625500988, −11.21997421576208337565995290066, −9.273312771289978362395807645424, −7.77298226708569010748330096188, −6.19787277208682350758293250680, −4.03233353353448226112015909150, 0.71653981401384729909148117373, 4.71778786158556375403026532737, 6.37121636747613915891305958555, 8.248567670133603496136051549933, 10.29577896762193454498923990624, 10.96124387966262686905612682403, 12.36492921282942078955564771609, 13.79486900884739976579409173593, 15.43812185960225948607248059498, 16.42249300166516496351507503810

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