Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.935 - 0.354i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16.4 − 5.99i)2-s + (7.02 + 46.2i)3-s + (136. + 114. i)4-s + (−40.0 + 227. i)5-s + (161. − 803. i)6-s + (882. − 740. i)7-s + (−444. − 770. i)8-s + (−2.08e3 + 649. i)9-s + (2.01e3 − 3.49e3i)10-s + (647. + 3.67e3i)11-s + (−4.35e3 + 7.14e3i)12-s + (−7.56e3 + 2.75e3i)13-s + (−1.89e4 + 6.90e3i)14-s + (−1.07e4 − 256. i)15-s + (−1.26e3 − 7.18e3i)16-s + (−1.07e4 + 1.86e4i)17-s + ⋯
L(s)  = 1  + (−1.45 − 0.529i)2-s + (0.150 + 0.988i)3-s + (1.07 + 0.897i)4-s + (−0.143 + 0.812i)5-s + (0.305 − 1.51i)6-s + (0.972 − 0.816i)7-s + (−0.307 − 0.532i)8-s + (−0.954 + 0.296i)9-s + (0.638 − 1.10i)10-s + (0.146 + 0.831i)11-s + (−0.726 + 1.19i)12-s + (−0.955 + 0.347i)13-s + (−1.84 + 0.672i)14-s + (−0.824 − 0.0196i)15-s + (−0.0773 − 0.438i)16-s + (−0.530 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.935 - 0.354i$
motivic weight  =  \(7\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :7/2),\ -0.935 - 0.354i)$
$L(4)$  $\approx$  $0.0699591 + 0.382067i$
$L(\frac12)$  $\approx$  $0.0699591 + 0.382067i$
$L(\frac{9}{2})$   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 \( 1 + (-7.02 - 46.2i)T \)
good2 \( 1 + (16.4 + 5.99i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (40.0 - 227. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (-882. + 740. i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (-647. - 3.67e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (7.56e3 - 2.75e3i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (1.07e4 - 1.86e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (5.56e3 + 9.63e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (7.96e4 + 6.68e4i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (1.32e5 + 4.82e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (1.01e4 + 8.48e3i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (2.03e5 - 3.51e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (1.21e5 - 4.42e4i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (-6.19e3 - 3.51e4i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (-8.14e5 + 6.83e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 - 6.66e5T + 1.17e12T^{2} \)
59 \( 1 + (2.93e5 - 1.66e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (1.87e6 - 1.57e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-3.89e6 + 1.41e6i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (2.19e6 - 3.80e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-1.13e6 - 1.95e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (7.23e6 + 2.63e6i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (1.87e6 + 6.83e5i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (3.29e6 + 5.70e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (-1.19e6 - 6.78e6i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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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

−16.85495452507466174422221580512, −15.14040329068764451349820403835, −14.26252221917717397793689078098, −11.72418665862497102154752131460, −10.64079072025136363780616489114, −10.04454434156900857470161747640, −8.559416339404853505186769305416, −7.27592534580110244319909421724, −4.37023778624955040980109782894, −2.18509208010014007201322171961, 0.29683285349390662424653429412, 1.83905621555347005715307107139, 5.61415168743287893104801981116, 7.43604432169968935278299012373, 8.375074566927182666211734945185, 9.239504616282323438054016084242, 11.29445808539946577223122844626, 12.46597320589544540970482559669, 14.12738941667177679279632663190, 15.58466823767755164110727981041

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