Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.363 - 0.931i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.69 + 3.52i)2-s + (−5.72 − 14.4i)3-s + (57.0 − 47.8i)4-s + (2.39 + 13.6i)5-s + (106. + 120. i)6-s + (−83.8 − 70.3i)7-s + (−218. + 379. i)8-s + (−177. + 166. i)9-s + (−71.2 − 123. i)10-s + (−96.8 + 549. i)11-s + (−1.02e3 − 552. i)12-s + (857. + 312. i)13-s + (1.06e3 + 386. i)14-s + (183. − 112. i)15-s + (370. − 2.10e3i)16-s + (−249. − 431. i)17-s + ⋯
L(s)  = 1  + (−1.71 + 0.623i)2-s + (−0.367 − 0.930i)3-s + (1.78 − 1.49i)4-s + (0.0429 + 0.243i)5-s + (1.20 + 1.36i)6-s + (−0.647 − 0.542i)7-s + (−1.20 + 2.09i)8-s + (−0.730 + 0.683i)9-s + (−0.225 − 0.390i)10-s + (−0.241 + 1.36i)11-s + (−2.04 − 1.10i)12-s + (1.40 + 0.512i)13-s + (1.44 + 0.526i)14-s + (0.210 − 0.129i)15-s + (0.362 − 2.05i)16-s + (−0.209 − 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.363 - 0.931i$
motivic weight  =  \(5\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :5/2),\ -0.363 - 0.931i)$
$L(3)$  $\approx$  $0.173059 + 0.253304i$
$L(\frac12)$  $\approx$  $0.173059 + 0.253304i$
$L(\frac{7}{2})$   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 \( 1 + (5.72 + 14.4i)T \)
good2 \( 1 + (9.69 - 3.52i)T + (24.5 - 20.5i)T^{2} \)
5 \( 1 + (-2.39 - 13.6i)T + (-2.93e3 + 1.06e3i)T^{2} \)
7 \( 1 + (83.8 + 70.3i)T + (2.91e3 + 1.65e4i)T^{2} \)
11 \( 1 + (96.8 - 549. i)T + (-1.51e5 - 5.50e4i)T^{2} \)
13 \( 1 + (-857. - 312. i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (249. + 431. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.43e3 - 2.47e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (296. - 249. i)T + (1.11e6 - 6.33e6i)T^{2} \)
29 \( 1 + (612. - 222. i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (4.64e3 - 3.89e3i)T + (4.97e6 - 2.81e7i)T^{2} \)
37 \( 1 + (-325. - 563. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (-3.37e3 - 1.22e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + (2.55e3 - 1.44e4i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (1.17e4 + 9.85e3i)T + (3.98e7 + 2.25e8i)T^{2} \)
53 \( 1 + 2.30e4T + 4.18e8T^{2} \)
59 \( 1 + (-275. - 1.56e3i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (2.31e3 + 1.94e3i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (6.27e3 + 2.28e3i)T + (1.03e9 + 8.67e8i)T^{2} \)
71 \( 1 + (-1.78e4 - 3.09e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (2.32e4 - 4.03e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.38e4 - 5.04e3i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (6.88e4 - 2.50e4i)T + (3.01e9 - 2.53e9i)T^{2} \)
89 \( 1 + (1.70e4 - 2.94e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (2.40e3 - 1.36e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−16.84810437038968786900109073514, −16.08836929225240419894249005066, −14.50210773078068208369064769656, −12.77083302299176946161784320490, −11.07650774151586278090228824256, −9.995208247883543688377127787651, −8.423000526501111399321740488111, −7.13501780986485790773824966343, −6.29008981487749903760322123526, −1.59984733964207088328073344245, 0.36689057539596026537878849194, 3.16582412336962924068589087086, 6.16520749361222063815554699036, 8.565399607990870991794505113325, 9.149273879428941154670263916873, 10.71415002126268946418234190593, 11.23676761747531666142267971250, 12.94250161588326111800478619137, 15.51444898388797411327320004297, 16.22339168204876822741817947810

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