Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.582 + 0.813i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.05 − 1.47i)2-s + (4.37 + 2.79i)3-s + (8.11 + 6.81i)4-s + (3.22 − 18.3i)5-s + (−13.6 − 17.7i)6-s + (14.4 − 12.0i)7-s + (−5.59 − 9.69i)8-s + (11.3 + 24.5i)9-s + (−40.0 + 69.4i)10-s + (−1.50 − 8.51i)11-s + (16.4 + 52.5i)12-s + (−3.90 + 1.42i)13-s + (−76.2 + 27.7i)14-s + (65.3 − 71.1i)15-s + (−6.33 − 35.9i)16-s + (−59.2 + 102. i)17-s + ⋯
L(s)  = 1  + (−1.43 − 0.521i)2-s + (0.842 + 0.538i)3-s + (1.01 + 0.851i)4-s + (0.288 − 1.63i)5-s + (−0.926 − 1.21i)6-s + (0.777 − 0.652i)7-s + (−0.247 − 0.428i)8-s + (0.420 + 0.907i)9-s + (−1.26 + 2.19i)10-s + (−0.0411 − 0.233i)11-s + (0.396 + 1.26i)12-s + (−0.0833 + 0.0303i)13-s + (−1.45 + 0.529i)14-s + (1.12 − 1.22i)15-s + (−0.0989 − 0.561i)16-s + (−0.845 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.582 + 0.813i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.582 + 0.813i)$
$L(2)$  $\approx$  $0.744504 - 0.382656i$
$L(\frac12)$  $\approx$  $0.744504 - 0.382656i$
$L(\frac{5}{2})$   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 \( 1 + (-4.37 - 2.79i)T \)
good2 \( 1 + (4.05 + 1.47i)T + (6.12 + 5.14i)T^{2} \)
5 \( 1 + (-3.22 + 18.3i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (-14.4 + 12.0i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (1.50 + 8.51i)T + (-1.25e3 + 455. i)T^{2} \)
13 \( 1 + (3.90 - 1.42i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (59.2 - 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.28 - 16.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (15.0 + 12.6i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (-189. - 68.8i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (25.8 + 21.6i)T + (5.17e3 + 2.93e4i)T^{2} \)
37 \( 1 + (26.0 - 45.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-89.8 + 32.6i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (-36.6 - 207. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (276. - 232. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + 126.T + 1.48e5T^{2} \)
59 \( 1 + (-123. + 702. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (-149. + 125. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (449. - 163. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-302. + 523. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-504. - 873. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (923. + 336. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (383. + 139. i)T + (4.38e5 + 3.67e5i)T^{2} \)
89 \( 1 + (-359. - 622. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (278. + 1.57e3i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−16.92525411702734617320392540071, −15.95017964347945755675977891616, −14.16643699850313981662086334751, −12.79581977129905413267465310437, −10.99749028270064734107793368988, −9.818882450311836723968877852037, −8.651948157776273597422872071686, −8.072977841897684627120108367686, −4.60610429253123592940544404010, −1.55119488685298694706377128376, 2.40021000513703018189083034113, 6.64739226322276257710649848641, 7.55464607637370912137995868645, 8.889624922750958048763327747118, 10.11078057509594653066416472527, 11.53790275536186566841490549802, 13.75291282782811139842126018861, 14.84391829593379355606655294640, 15.68283288741570199198727831507, 17.83209483177300729669361943477

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