Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.919 - 0.392i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 + 1.17i)2-s + (4.34 − 2.84i)3-s + (−0.812 + 4.61i)4-s + (12.7 − 4.62i)5-s + (−2.72 + 9.06i)6-s + (3.96 + 22.5i)7-s + (−11.5 − 20.0i)8-s + (10.7 − 24.7i)9-s + (−12.3 + 21.3i)10-s + (−31.0 − 11.3i)11-s + (9.60 + 22.3i)12-s + (−52.2 − 43.8i)13-s + (−31.8 − 26.7i)14-s + (42.0 − 56.3i)15-s + (4.35 + 1.58i)16-s + (−18.6 + 32.2i)17-s + ⋯
L(s)  = 1  + (−0.493 + 0.414i)2-s + (0.836 − 0.548i)3-s + (−0.101 + 0.576i)4-s + (1.13 − 0.413i)5-s + (−0.185 + 0.616i)6-s + (0.214 + 1.21i)7-s + (−0.510 − 0.884i)8-s + (0.398 − 0.917i)9-s + (−0.389 + 0.674i)10-s + (−0.852 − 0.310i)11-s + (0.231 + 0.537i)12-s + (−1.11 − 0.935i)13-s + (−0.608 − 0.510i)14-s + (0.723 − 0.969i)15-s + (0.0680 + 0.0247i)16-s + (−0.265 + 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.919 - 0.392i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.919 - 0.392i)$
$L(2)$  $\approx$  $1.21136 + 0.247664i$
$L(\frac12)$  $\approx$  $1.21136 + 0.247664i$
$L(\frac{5}{2})$   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 \( 1 + (-4.34 + 2.84i)T \)
good2 \( 1 + (1.39 - 1.17i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (-12.7 + 4.62i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-3.96 - 22.5i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (31.0 + 11.3i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (52.2 + 43.8i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (18.6 - 32.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-47.5 - 82.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-16.6 + 94.3i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (109. - 92.1i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-13.9 + 79.0i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-32.8 + 56.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (75.7 + 63.5i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-413. - 150. i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-46.4 - 263. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 38.9T + 1.48e5T^{2} \)
59 \( 1 + (202. - 73.6i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (9.24 + 52.4i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-793. - 665. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (148. - 256. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-49.0 - 84.9i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (468. - 392. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-657. + 552. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (663. + 1.14e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (631. + 229. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−17.21667287585800045124515276432, −15.73173866515925514226922765006, −14.52264023902250018165270808624, −12.98474929791622173262833070957, −12.44613565856752611549917810722, −9.788717066626284839446534854869, −8.745832460973875162139288880515, −7.70716186055707683963993584616, −5.76551226002255136377169438777, −2.58614205665582351900876399237, 2.26729969812163849923429875686, 4.95645819037423519360715457586, 7.31429534123810868959005234221, 9.366083624223471638927266697836, 9.988933121243560212036657527287, 11.02198741519528267478134274096, 13.62016856406472391721261290818, 14.08259270305385614809249345394, 15.33475096739504984960096620618, 17.04731410986042419436498870501

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