Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.213 + 0.976i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.58 + 3.00i)2-s + (−2.57 − 4.51i)3-s + (2.41 − 13.7i)4-s + (2.94 − 1.07i)5-s + (22.8 + 8.45i)6-s + (−5.29 − 30.0i)7-s + (13.8 + 23.9i)8-s + (−13.7 + 23.2i)9-s + (−7.33 + 12.7i)10-s + (−49.0 − 17.8i)11-s + (−68.0 + 24.3i)12-s + (11.6 + 9.79i)13-s + (109. + 91.7i)14-s + (−12.4 − 10.5i)15-s + (−17.1 − 6.24i)16-s + (35.3 − 61.1i)17-s + ⋯
L(s)  = 1  + (−1.26 + 1.06i)2-s + (−0.494 − 0.868i)3-s + (0.301 − 1.71i)4-s + (0.263 − 0.0958i)5-s + (1.55 + 0.575i)6-s + (−0.286 − 1.62i)7-s + (0.611 + 1.05i)8-s + (−0.509 + 0.860i)9-s + (−0.231 + 0.401i)10-s + (−1.34 − 0.489i)11-s + (−1.63 + 0.585i)12-s + (0.248 + 0.208i)13-s + (2.08 + 1.75i)14-s + (−0.213 − 0.181i)15-s + (−0.268 − 0.0975i)16-s + (0.503 − 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.213 + 0.976i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.213 + 0.976i)$
$L(2)$  $\approx$  $0.317000 - 0.255142i$
$L(\frac12)$  $\approx$  $0.317000 - 0.255142i$
$L(\frac{5}{2})$   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 \( 1 + (2.57 + 4.51i)T \)
good2 \( 1 + (3.58 - 3.00i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (-2.94 + 1.07i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (5.29 + 30.0i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (49.0 + 17.8i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (-11.6 - 9.79i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-35.3 + 61.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-32.0 - 55.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-9.07 + 51.4i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-20.2 + 17.0i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-18.8 + 106. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-175. + 303. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-147. - 123. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-132. - 48.1i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (20.0 + 113. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 3.24T + 1.48e5T^{2} \)
59 \( 1 + (-725. + 264. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-32.0 - 181. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (349. + 293. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-88.2 + 152. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-23.6 - 40.9i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (240. - 201. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (36.6 - 30.7i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (306. + 531. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.04e3 + 380. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−16.64339915878723322822907007547, −16.14988785196739798552304829085, −14.12563817428992292439003912557, −13.10925020067327030773555178038, −10.97129756328046289531242356890, −9.876976151195152515513810591516, −7.949952726329945449819666474017, −7.22287331782653972248767262084, −5.77041627029597217245968415408, −0.63009003674638723270792936967, 2.75330951754792471959701340071, 5.59662149789716191823662619521, 8.357222078817441183572035732107, 9.541951115687385947375546575074, 10.41171236584175693236762468334, 11.65968868541570027926024902029, 12.66784195002551794993751928511, 15.17705642150260768904107141461, 16.07513193046817669875321460518, 17.61287986536980364543541890215

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