Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.0952 + 0.995i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.9 − 5.08i)2-s + (−44.6 + 13.9i)3-s + (71.4 + 59.9i)4-s + (−48.7 + 276. i)5-s + (694. + 31.8i)6-s + (−1.11e3 + 934. i)7-s + (258. + 447. i)8-s + (1.79e3 − 1.24e3i)9-s + (2.08e3 − 3.61e3i)10-s + (−907. − 5.14e3i)11-s + (−4.02e3 − 1.67e3i)12-s + (−1.22e4 + 4.45e3i)13-s + (2.03e4 − 7.39e3i)14-s + (−1.68e3 − 1.30e4i)15-s + (−3.40e3 − 1.93e4i)16-s + (9.92e3 − 1.71e4i)17-s + ⋯
L(s)  = 1  + (−1.23 − 0.449i)2-s + (−0.954 + 0.298i)3-s + (0.558 + 0.468i)4-s + (−0.174 + 0.989i)5-s + (1.31 + 0.0601i)6-s + (−1.22 + 1.02i)7-s + (0.178 + 0.308i)8-s + (0.821 − 0.570i)9-s + (0.660 − 1.14i)10-s + (−0.205 − 1.16i)11-s + (−0.672 − 0.280i)12-s + (−1.54 + 0.562i)13-s + (1.97 − 0.720i)14-s + (−0.129 − 0.996i)15-s + (−0.207 − 1.17i)16-s + (0.489 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.0952 + 0.995i$
motivic weight  =  \(7\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :7/2),\ 0.0952 + 0.995i)$
$L(4)$  $\approx$  $0.124841 - 0.113468i$
$L(\frac12)$  $\approx$  $0.124841 - 0.113468i$
$L(\frac{9}{2})$   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 \( 1 + (44.6 - 13.9i)T \)
good2 \( 1 + (13.9 + 5.08i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (48.7 - 276. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (1.11e3 - 934. i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (907. + 5.14e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (1.22e4 - 4.45e3i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (-9.92e3 + 1.71e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.63e4 - 2.82e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (892. + 749. i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (-1.14e5 - 4.17e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (1.35e5 + 1.13e5i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (-2.09e5 + 3.62e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (-2.15e5 + 7.82e4i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (2.19e4 + 1.24e5i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (3.25e5 - 2.72e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 - 6.51e5T + 1.17e12T^{2} \)
59 \( 1 + (-3.13e4 + 1.77e5i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (1.87e6 - 1.57e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-6.34e5 + 2.30e5i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (-9.91e4 + 1.71e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-1.58e6 - 2.74e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (1.88e6 + 6.86e5i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (5.41e6 + 1.97e6i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (3.84e6 + 6.65e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (-6.44e5 - 3.65e6i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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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

−16.04944656616964756038219403793, −14.44842902346729645952531047492, −12.28812069447279424215222007018, −11.32456622179595813000835112255, −10.10481798653275633926258431654, −9.300135421737245063457782473448, −7.24652738853338937844538214513, −5.68214990621113184623133161165, −2.84378147389782381592393657134, −0.20177873315153876511001413273, 0.888633883071739300657557951584, 4.70128361722649610811235205287, 6.80648442253801444258952391015, 7.71063055971530692767323266676, 9.646552748661854670907449998973, 10.30179391215914618249013507220, 12.42926084108595331221758015444, 13.04273281290735780803491417464, 15.53892166355737679752982091909, 16.63584575122103206069527331586

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