Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.798 + 0.602i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.26 − 1.89i)2-s + (5.05 − 1.18i)3-s + (0.123 − 0.703i)4-s + (−16.0 + 5.82i)5-s + (9.18 − 12.2i)6-s + (−2.22 − 12.6i)7-s + (10.7 + 18.6i)8-s + (24.1 − 12.0i)9-s + (−25.1 + 43.5i)10-s + (−48.2 − 17.5i)11-s + (−0.207 − 3.70i)12-s + (42.4 + 35.6i)13-s + (−29.0 − 24.3i)14-s + (−74.0 + 48.4i)15-s + (65.0 + 23.6i)16-s + (18.6 − 32.2i)17-s + ⋯
L(s)  = 1  + (0.799 − 0.670i)2-s + (0.973 − 0.228i)3-s + (0.0154 − 0.0879i)4-s + (−1.43 + 0.521i)5-s + (0.625 − 0.835i)6-s + (−0.120 − 0.682i)7-s + (0.475 + 0.823i)8-s + (0.895 − 0.444i)9-s + (−0.795 + 1.37i)10-s + (−1.32 − 0.481i)11-s + (−0.00499 − 0.0891i)12-s + (0.906 + 0.760i)13-s + (−0.553 − 0.464i)14-s + (−1.27 + 0.834i)15-s + (1.01 + 0.369i)16-s + (0.265 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.798 + 0.602i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.798 + 0.602i)$
$L(2)$  $\approx$  $1.70057 - 0.569866i$
$L(\frac12)$  $\approx$  $1.70057 - 0.569866i$
$L(\frac{5}{2})$   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 \( 1 + (-5.05 + 1.18i)T \)
good2 \( 1 + (-2.26 + 1.89i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (16.0 - 5.82i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (2.22 + 12.6i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (48.2 + 17.5i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (-42.4 - 35.6i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-18.6 + 32.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (3.43 + 5.94i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-5.91 + 33.5i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (78.5 - 65.8i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-38.1 + 216. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (159. - 275. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (77.7 + 65.2i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (97.2 + 35.3i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-85.0 - 482. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 - 136.T + 1.48e5T^{2} \)
59 \( 1 + (-195. + 71.1i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-103. - 587. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (336. + 282. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (69.9 - 121. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (213. + 370. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (555. - 466. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-589. + 494. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-371. - 642. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.28e3 + 466. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−16.35212924636990458857818222690, −15.23801066179166500263777283372, −13.93191131333126109834920879659, −13.13960456117214986645442527731, −11.75042050531947149249127609003, −10.65197439708208584226565117417, −8.339547171840579891679976874018, −7.36517375460138514240794208246, −4.17218920404259020432204081166, −3.07458366110846014994426482639, 3.68381314538710594896729655739, 5.21378327244305811766685806340, 7.50690021053379772851534056314, 8.542448590438372136752126157050, 10.40892788285278227939266349365, 12.46839986442950781521692037345, 13.32861838424103976944873382124, 14.89561990588879921356640289779, 15.61422307208800208590287224928, 16.02450731723506500100630738115

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