Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.526 + 0.850i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.79 − 3.20i)2-s + (13.9 − 6.86i)3-s + (42.5 + 35.7i)4-s + (−11.0 + 62.9i)5-s + (−145. + 15.6i)6-s + (158. − 133. i)7-s + (−110. − 191. i)8-s + (148. − 192. i)9-s + (299. − 518. i)10-s + (−38.4 − 218. i)11-s + (841. + 207. i)12-s + (10.0 − 3.65i)13-s + (−1.82e3 + 662. i)14-s + (276. + 957. i)15-s + (49.9 + 283. i)16-s + (869. − 1.50e3i)17-s + ⋯
L(s)  = 1  + (−1.55 − 0.565i)2-s + (0.897 − 0.440i)3-s + (1.33 + 1.11i)4-s + (−0.198 + 1.12i)5-s + (−1.64 + 0.177i)6-s + (1.22 − 1.02i)7-s + (−0.610 − 1.05i)8-s + (0.611 − 0.791i)9-s + (0.945 − 1.63i)10-s + (−0.0958 − 0.543i)11-s + (1.68 + 0.416i)12-s + (0.0165 − 0.00600i)13-s + (−2.48 + 0.903i)14-s + (0.317 + 1.09i)15-s + (0.0488 + 0.276i)16-s + (0.729 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.526 + 0.850i$
motivic weight  =  \(5\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :5/2),\ 0.526 + 0.850i)$
$L(3)$  $\approx$  $0.909658 - 0.506747i$
$L(\frac12)$  $\approx$  $0.909658 - 0.506747i$
$L(\frac{7}{2})$   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 \( 1 + (-13.9 + 6.86i)T \)
good2 \( 1 + (8.79 + 3.20i)T + (24.5 + 20.5i)T^{2} \)
5 \( 1 + (11.0 - 62.9i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (-158. + 133. i)T + (2.91e3 - 1.65e4i)T^{2} \)
11 \( 1 + (38.4 + 218. i)T + (-1.51e5 + 5.50e4i)T^{2} \)
13 \( 1 + (-10.0 + 3.65i)T + (2.84e5 - 2.38e5i)T^{2} \)
17 \( 1 + (-869. + 1.50e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.05e3 - 1.82e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.06e3 - 1.73e3i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (2.50e3 + 913. i)T + (1.57e7 + 1.31e7i)T^{2} \)
31 \( 1 + (4.66e3 + 3.91e3i)T + (4.97e6 + 2.81e7i)T^{2} \)
37 \( 1 + (2.83e3 - 4.90e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (-7.53e3 + 2.74e3i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (-1.91e3 - 1.08e4i)T + (-1.38e8 + 5.02e7i)T^{2} \)
47 \( 1 + (9.34e3 - 7.83e3i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 + 1.86e4T + 4.18e8T^{2} \)
59 \( 1 + (981. - 5.56e3i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (3.76e4 - 3.15e4i)T + (1.46e8 - 8.31e8i)T^{2} \)
67 \( 1 + (5.41e4 - 1.97e4i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-1.46e4 + 2.53e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (2.17e3 + 3.76e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.20e4 - 4.38e3i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (-5.35e4 - 1.94e4i)T + (3.01e9 + 2.53e9i)T^{2} \)
89 \( 1 + (-1.74e4 - 3.01e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (1.69e3 + 9.62e3i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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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

−16.50894605552076182663976673655, −14.71953476160308789697499115424, −13.84015266280042015788957110211, −11.64651844846385179601087562976, −10.72720146754200629166581827516, −9.467359731515441999638754620480, −7.80179024597256774560507465066, −7.40390898027986391925625303052, −3.15496720853153792086454281074, −1.28624085030385828212866407526, 1.65455893594982054600148221959, 5.02479798297097771641379861486, 7.64996757459071724047026195888, 8.635896484905596094477349794042, 9.213150301076596931270007327783, 10.82274055884332767472212312203, 12.62090998557436477669753659616, 14.69826300684777572436978380465, 15.48229910866273758689944337001, 16.55168069283171831229962761286

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