Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.0270 + 0.999i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.97 + 1.81i)2-s + (−3.67 + 3.67i)3-s + (15.3 − 12.9i)4-s + (−1.82 − 10.3i)5-s + (11.6 − 24.9i)6-s + (−5.92 − 4.96i)7-s + (−31.9 + 55.3i)8-s + (0.0364 − 26.9i)9-s + (27.8 + 48.2i)10-s + (2.99 − 16.9i)11-s + (−9.15 + 103. i)12-s + (−67.4 − 24.5i)13-s + (38.4 + 14.0i)14-s + (44.7 + 31.3i)15-s + (30.9 − 175. i)16-s + (−23.3 − 40.4i)17-s + ⋯
L(s)  = 1  + (−1.76 + 0.640i)2-s + (−0.707 + 0.706i)3-s + (1.92 − 1.61i)4-s + (−0.163 − 0.925i)5-s + (0.792 − 1.69i)6-s + (−0.319 − 0.268i)7-s + (−1.41 + 2.44i)8-s + (0.00134 − 0.999i)9-s + (0.880 + 1.52i)10-s + (0.0820 − 0.465i)11-s + (−0.220 + 2.49i)12-s + (−1.43 − 0.523i)13-s + (0.734 + 0.267i)14-s + (0.769 + 0.539i)15-s + (0.483 − 2.74i)16-s + (−0.333 − 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0270 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0270 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.0270 + 0.999i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.0270 + 0.999i)$
$L(2)$  $\approx$  $0.142034 - 0.138241i$
$L(\frac12)$  $\approx$  $0.142034 - 0.138241i$
$L(\frac{5}{2})$   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 \( 1 + (3.67 - 3.67i)T \)
good2 \( 1 + (4.97 - 1.81i)T + (6.12 - 5.14i)T^{2} \)
5 \( 1 + (1.82 + 10.3i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (5.92 + 4.96i)T + (59.5 + 337. i)T^{2} \)
11 \( 1 + (-2.99 + 16.9i)T + (-1.25e3 - 455. i)T^{2} \)
13 \( 1 + (67.4 + 24.5i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (23.3 + 40.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (21.3 - 36.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36.3 + 30.5i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (139. - 50.8i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (120. - 101. i)T + (5.17e3 - 2.93e4i)T^{2} \)
37 \( 1 + (-35.3 - 61.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (219. + 79.8i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (-71.5 + 405. i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (-327. - 275. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 - 43.9T + 1.48e5T^{2} \)
59 \( 1 + (57.9 + 328. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-292. - 245. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (230. + 83.9i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (-403. - 698. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-175. + 303. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-9.99e2 + 363. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (959. - 349. i)T + (4.38e5 - 3.67e5i)T^{2} \)
89 \( 1 + (-320. + 555. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-60.5 + 343. i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−16.75643505204580486783296742661, −15.98113181031536309994430315961, −14.85529326700521138282685391270, −12.26822145856954319916109403850, −10.82066414227898608831399278589, −9.736481879280681309870817747118, −8.720406020343865834160935969782, −7.06965634178584925744342632286, −5.34841752229001997160381366933, −0.34709593077116122194215721108, 2.30508093884605609361262026822, 6.73368558866435084205526898646, 7.60684881628411948000616370884, 9.434471499082734571318383716230, 10.69664984093828308744753164082, 11.62975354132174677959807218031, 12.71722549467447083248042933156, 15.08042784199882822683271098189, 16.67368131288198248801047995580, 17.42554496482041569484900563876

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