Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.294 + 0.955i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 + 0.772i)2-s + (−0.789 − 5.13i)3-s + (−2.22 + 1.86i)4-s + (−3.33 − 18.9i)5-s + (5.64 + 10.2i)6-s + (0.500 + 0.420i)7-s + (12.3 − 21.3i)8-s + (−25.7 + 8.11i)9-s + (21.7 + 37.6i)10-s + (−4.24 + 24.0i)11-s + (11.3 + 9.94i)12-s + (36.6 + 13.3i)13-s + (−1.38 − 0.504i)14-s + (−94.6 + 32.1i)15-s + (−5.61 + 31.8i)16-s + (−20.0 − 34.6i)17-s + ⋯
L(s)  = 1  + (−0.750 + 0.273i)2-s + (−0.151 − 0.988i)3-s + (−0.277 + 0.233i)4-s + (−0.298 − 1.69i)5-s + (0.383 + 0.699i)6-s + (0.0270 + 0.0226i)7-s + (0.543 − 0.942i)8-s + (−0.953 + 0.300i)9-s + (0.686 + 1.18i)10-s + (−0.116 + 0.659i)11-s + (0.272 + 0.239i)12-s + (0.781 + 0.284i)13-s + (−0.0264 − 0.00963i)14-s + (−1.62 + 0.552i)15-s + (−0.0878 + 0.497i)16-s + (−0.285 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.294 + 0.955i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.294 + 0.955i)$
$L(2)$  $\approx$  $0.354687 - 0.480321i$
$L(\frac12)$  $\approx$  $0.354687 - 0.480321i$
$L(\frac{5}{2})$   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 \( 1 + (0.789 + 5.13i)T \)
good2 \( 1 + (2.12 - 0.772i)T + (6.12 - 5.14i)T^{2} \)
5 \( 1 + (3.33 + 18.9i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (-0.500 - 0.420i)T + (59.5 + 337. i)T^{2} \)
11 \( 1 + (4.24 - 24.0i)T + (-1.25e3 - 455. i)T^{2} \)
13 \( 1 + (-36.6 - 13.3i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (20.0 + 34.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-76.3 + 132. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-89.8 + 75.4i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (83.5 - 30.4i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (-58.9 + 49.4i)T + (5.17e3 - 2.93e4i)T^{2} \)
37 \( 1 + (-108. - 187. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (48.1 + 17.5i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (-26.9 + 152. i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (-100. - 84.2i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + 362.T + 1.48e5T^{2} \)
59 \( 1 + (-60.3 - 342. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-279. - 234. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-749. - 272. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (185. + 320. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (253. - 438. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (143. - 52.1i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (-457. + 166. i)T + (4.38e5 - 3.67e5i)T^{2} \)
89 \( 1 + (-44.9 + 77.8i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-197. + 1.11e3i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−16.80132581173430601254384319917, −15.79398171338706201525471537378, −13.45806776849141388411865343839, −12.87348505311779030933913625672, −11.62121803666297794156862900784, −9.242262122640349428930311305613, −8.438186392234704453265647420931, −7.14432603841726370368899543670, −4.84347307980598724335577087435, −0.825891913404535097636267097145, 3.47578625177319230596792451407, 5.86901509252705996686486337698, 8.042980729264788706290229597012, 9.645973874038122501382679479620, 10.72475569097974574075323535154, 11.27211842028167878124929406706, 13.96404769714766129064834373138, 14.79396146210322405645640898399, 15.98625603577678809018667325972, 17.45464229534682582317229265412

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