Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.947 - 0.319i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.25 + 1.18i)2-s + (−12.0 + 9.88i)3-s + (−15.3 − 12.8i)4-s + (−5.36 + 30.4i)5-s + (−50.9 + 17.8i)6-s + (−155. + 130. i)7-s + (−90.0 − 155. i)8-s + (47.6 − 238. i)9-s + (−53.5 + 92.7i)10-s + (70.5 + 400. i)11-s + (311. + 3.55i)12-s + (235. − 85.8i)13-s + (−661. + 240. i)14-s + (−236. − 420. i)15-s + (2.89 + 16.3i)16-s + (−8.80 + 15.2i)17-s + ⋯
L(s)  = 1  + (0.575 + 0.209i)2-s + (−0.773 + 0.634i)3-s + (−0.478 − 0.401i)4-s + (−0.0960 + 0.544i)5-s + (−0.577 + 0.202i)6-s + (−1.20 + 1.00i)7-s + (−0.497 − 0.861i)8-s + (0.196 − 0.980i)9-s + (−0.169 + 0.293i)10-s + (0.175 + 0.997i)11-s + (0.625 + 0.00712i)12-s + (0.387 − 0.140i)13-s + (−0.902 + 0.328i)14-s + (−0.271 − 0.482i)15-s + (0.00282 + 0.0160i)16-s + (−0.00738 + 0.0127i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.947 - 0.319i$
motivic weight  =  \(5\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :5/2),\ -0.947 - 0.319i)$
$L(3)$  $\approx$  $0.107216 + 0.653780i$
$L(\frac12)$  $\approx$  $0.107216 + 0.653780i$
$L(\frac{7}{2})$   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 \( 1 + (12.0 - 9.88i)T \)
good2 \( 1 + (-3.25 - 1.18i)T + (24.5 + 20.5i)T^{2} \)
5 \( 1 + (5.36 - 30.4i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (155. - 130. i)T + (2.91e3 - 1.65e4i)T^{2} \)
11 \( 1 + (-70.5 - 400. i)T + (-1.51e5 + 5.50e4i)T^{2} \)
13 \( 1 + (-235. + 85.8i)T + (2.84e5 - 2.38e5i)T^{2} \)
17 \( 1 + (8.80 - 15.2i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-977. - 1.69e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.70e3 + 1.42e3i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (7.23e3 + 2.63e3i)T + (1.57e7 + 1.31e7i)T^{2} \)
31 \( 1 + (3.42e3 + 2.86e3i)T + (4.97e6 + 2.81e7i)T^{2} \)
37 \( 1 + (5.00e3 - 8.66e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (8.92e3 - 3.24e3i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (-2.49e3 - 1.41e4i)T + (-1.38e8 + 5.02e7i)T^{2} \)
47 \( 1 + (-8.65e3 + 7.26e3i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 - 2.63e4T + 4.18e8T^{2} \)
59 \( 1 + (946. - 5.36e3i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (-2.98e4 + 2.50e4i)T + (1.46e8 - 8.31e8i)T^{2} \)
67 \( 1 + (3.77e4 - 1.37e4i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-4.60e3 + 7.97e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-1.94e4 - 3.37e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.23e4 - 1.17e4i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (-9.43e4 - 3.43e4i)T + (3.01e9 + 2.53e9i)T^{2} \)
89 \( 1 + (5.38e4 + 9.31e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (-1.93e4 - 1.09e5i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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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

−16.53394894798674977398459548211, −15.40777568700096616117093032380, −14.75690148718010148568983534567, −12.97826370547742547387180125003, −11.97351589478117846145065539836, −10.17527323267207377637693916757, −9.353125234543568868944652271156, −6.53823465581045354504896975940, −5.50282555620141508778652325540, −3.71346876716643003650845342900, 0.40076860275539670332861974446, 3.68864276463237222434953188872, 5.50523649057887931648763813719, 7.17643566146626267368565084560, 8.968743986288062724475890815598, 10.89380914771104339218957912322, 12.20970449436681802884700386999, 13.25873981047629167654338752795, 13.77201003628963650298355391544, 16.20437513662070896496441773029

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