Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.249 + 0.968i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.90 + 3.60i)2-s + (7.82 − 46.1i)3-s + (−13.0 − 10.9i)4-s + (−29.9 + 170. i)5-s + (243. − 428. i)6-s + (1.22e3 − 1.02e3i)7-s + (−763. − 1.32e3i)8-s + (−2.06e3 − 721. i)9-s + (−909. + 1.57e3i)10-s + (−790. − 4.48e3i)11-s + (−605. + 514. i)12-s + (1.00e3 − 365. i)13-s + (1.58e4 − 5.75e3i)14-s + (7.60e3 + 2.71e3i)15-s + (−2.41e3 − 1.37e4i)16-s + (−3.55e3 + 6.15e3i)17-s + ⋯
L(s)  = 1  + (0.875 + 0.318i)2-s + (0.167 − 0.985i)3-s + (−0.101 − 0.0852i)4-s + (−0.107 + 0.608i)5-s + (0.460 − 0.809i)6-s + (1.34 − 1.13i)7-s + (−0.527 − 0.913i)8-s + (−0.943 − 0.330i)9-s + (−0.287 + 0.498i)10-s + (−0.179 − 1.01i)11-s + (−0.101 + 0.0859i)12-s + (0.126 − 0.0460i)13-s + (1.54 − 0.560i)14-s + (0.581 + 0.207i)15-s + (−0.147 − 0.836i)16-s + (−0.175 + 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.249 + 0.968i$
motivic weight  =  \(7\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :7/2),\ 0.249 + 0.968i)$
$L(4)$  $\approx$  $1.99419 - 1.54618i$
$L(\frac12)$  $\approx$  $1.99419 - 1.54618i$
$L(\frac{9}{2})$   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 \( 1 + (-7.82 + 46.1i)T \)
good2 \( 1 + (-9.90 - 3.60i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (29.9 - 170. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (-1.22e3 + 1.02e3i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (790. + 4.48e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (-1.00e3 + 365. i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (3.55e3 - 6.15e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-2.87e4 - 4.98e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-8.84e3 - 7.41e3i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (8.06e3 + 2.93e3i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (-1.07e5 - 9.04e4i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (-1.62e5 + 2.81e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (7.92e5 - 2.88e5i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (8.09e4 + 4.58e5i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (-5.59e5 + 4.69e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 - 1.20e6T + 1.17e12T^{2} \)
59 \( 1 + (3.03e5 - 1.72e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (-4.16e4 + 3.49e4i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-5.55e5 + 2.02e5i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (-7.25e5 + 1.25e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-2.04e6 - 3.54e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (8.73e5 + 3.17e5i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (-4.23e6 - 1.54e6i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (2.08e6 + 3.60e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (6.85e4 + 3.88e5i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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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

−14.88778144781865332021841361085, −14.07035094392189873360552457470, −13.51177812006705462576035791736, −11.89926285156279669035012920286, −10.58556112341766596161718350669, −8.270208496464527505304454315927, −7.02781999301935841190198371413, −5.52620206135496971664166384109, −3.60598948144728576213065946630, −1.07866231344716774191599262167, 2.58292375514178855854865517783, 4.66951758777794297151536959586, 5.11848918107179600197192188815, 8.297980861506861045281358928220, 9.263417115558557300517806144328, 11.29463120899427486047965345932, 12.10434529941603220024785748590, 13.60432832497131039507240654309, 14.86380115483490887170620683269, 15.52474225160572330248121646084

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