Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.778 - 0.628i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.874 + 4.96i)2-s + (2.71 + 4.43i)3-s + (−16.3 − 5.94i)4-s + (11.9 − 10.0i)5-s + (−24.3 + 9.58i)6-s + (−5.29 + 1.92i)7-s + (23.6 − 40.8i)8-s + (−12.2 + 24.0i)9-s + (39.2 + 67.9i)10-s + (30.2 + 25.3i)11-s + (−17.9 − 88.4i)12-s + (−10.1 − 57.7i)13-s + (−4.93 − 27.9i)14-s + (76.7 + 25.7i)15-s + (75.7 + 63.5i)16-s + (8.53 + 14.7i)17-s + ⋯
L(s)  = 1  + (−0.309 + 1.75i)2-s + (0.522 + 0.852i)3-s + (−2.04 − 0.742i)4-s + (1.06 − 0.895i)5-s + (−1.65 + 0.651i)6-s + (−0.286 + 0.104i)7-s + (1.04 − 1.80i)8-s + (−0.454 + 0.890i)9-s + (1.24 + 2.14i)10-s + (0.828 + 0.695i)11-s + (−0.432 − 2.12i)12-s + (−0.217 − 1.23i)13-s + (−0.0941 − 0.533i)14-s + (1.32 + 0.442i)15-s + (1.18 + 0.992i)16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.778 - 0.628i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.778 - 0.628i)$
$L(2)$  $\approx$  $0.391189 + 1.10742i$
$L(\frac12)$  $\approx$  $0.391189 + 1.10742i$
$L(\frac{5}{2})$   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 \( 1 + (-2.71 - 4.43i)T \)
good2 \( 1 + (0.874 - 4.96i)T + (-7.51 - 2.73i)T^{2} \)
5 \( 1 + (-11.9 + 10.0i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (5.29 - 1.92i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (-30.2 - 25.3i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (10.1 + 57.7i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (-8.53 - 14.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.5 + 18.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (27.3 + 9.97i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-34.8 + 197. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-36.3 - 13.2i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (144. + 250. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-78.5 - 445. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (12.7 + 10.6i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (101. - 36.9i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + 540.T + 1.48e5T^{2} \)
59 \( 1 + (-124. + 104. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (425. - 155. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (-53.9 - 306. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (-207. - 360. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-294. + 509. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-81.7 + 463. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-43.8 + 248. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (468. - 811. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-247. - 207. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−17.13216668283388346578305357412, −16.10603143303662692080668794587, −15.11620907433653858600388710557, −14.11760383557708974374869025152, −12.97568939297595617399632422017, −9.903968027292305536486948093785, −9.241862940332867323277880676873, −7.992956626194776035833065474463, −6.00863023957846445662444521159, −4.77453529998420347210767179716, 1.77408341863338498162756458293, 3.28732383405601425324092237854, 6.62224568916212795878911709631, 8.878690912385596378792260597043, 9.859844417616106623595259230175, 11.27515522777005490094028171080, 12.39547213234928021911300926277, 13.82799258253065038106255469527, 14.15342246495863850454010684125, 17.09381183124408818710922200734

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