Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.674 - 0.738i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.7 + 6.11i)2-s + (46.7 − 1.15i)3-s + (146. + 122. i)4-s + (−21.3 + 121. i)5-s + (792. + 266. i)6-s + (−285. + 239. i)7-s + (565. + 980. i)8-s + (2.18e3 − 107. i)9-s + (−1.10e3 + 1.90e3i)10-s + (−803. − 4.55e3i)11-s + (6.99e3 + 5.58e3i)12-s + (933. − 339. i)13-s + (−6.26e3 + 2.27e3i)14-s + (−860. + 5.69e3i)15-s + (−740. − 4.20e3i)16-s + (−1.11e4 + 1.93e4i)17-s + ⋯
L(s)  = 1  + (1.48 + 0.540i)2-s + (0.999 − 0.0246i)3-s + (1.14 + 0.960i)4-s + (−0.0765 + 0.434i)5-s + (1.49 + 0.503i)6-s + (−0.314 + 0.264i)7-s + (0.390 + 0.676i)8-s + (0.998 − 0.0493i)9-s + (−0.348 + 0.602i)10-s + (−0.182 − 1.03i)11-s + (1.16 + 0.932i)12-s + (0.117 − 0.0428i)13-s + (−0.609 + 0.221i)14-s + (−0.0657 + 0.435i)15-s + (−0.0452 − 0.256i)16-s + (−0.551 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.674 - 0.738i$
motivic weight  =  \(7\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :7/2),\ 0.674 - 0.738i)$
$L(4)$  $\approx$  $4.16168 + 1.83444i$
$L(\frac12)$  $\approx$  $4.16168 + 1.83444i$
$L(\frac{9}{2})$   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 \( 1 + (-46.7 + 1.15i)T \)
good2 \( 1 + (-16.7 - 6.11i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (21.3 - 121. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (285. - 239. i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (803. + 4.55e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (-933. + 339. i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (1.11e4 - 1.93e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (1.38e4 + 2.40e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-2.33e4 - 1.95e4i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (1.46e5 + 5.33e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (2.45e5 + 2.06e5i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (8.81e4 - 1.52e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (-4.56e5 + 1.66e5i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (-1.09e5 - 6.19e5i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (7.84e4 - 6.58e4i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 - 2.07e6T + 1.17e12T^{2} \)
59 \( 1 + (4.42e5 - 2.50e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (1.00e6 - 8.40e5i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (2.45e6 - 8.93e5i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (-4.29e5 + 7.44e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-1.52e6 - 2.64e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (6.42e5 + 2.33e5i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (-1.46e6 - 5.35e5i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (1.37e6 + 2.37e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (-1.45e5 - 8.26e5i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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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

−15.30132677129067448826941379071, −14.76466525090176963780133654133, −13.45744595197499751759356099475, −12.88635844697360454148219730108, −11.05650035963131398495521214870, −9.008853407441475344111587443076, −7.35135119755347123621473730370, −5.95693250709098969585533285034, −4.01675366444300590245605743438, −2.77697507314898640161983936999, 2.06757174052282325825117870834, 3.65607764552942645942855083294, 4.92298174544735185252840111452, 7.08089892400597355958727807096, 9.013330030804004113969920547073, 10.61597209672876433992127259858, 12.40552005408849694359944326463, 13.03537712033033962574315688659, 14.21175962206584853244343222014, 15.06690197187789764474324189667

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