Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.921 + 0.388i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.752 + 0.273i)2-s + (−2.15 + 15.4i)3-s + (−24.0 + 20.1i)4-s + (−15.6 − 88.6i)5-s + (−2.60 − 12.2i)6-s + (−70.2 − 58.9i)7-s + (25.3 − 43.9i)8-s + (−233. − 66.4i)9-s + (36.0 + 62.4i)10-s + (−109. + 623. i)11-s + (−259. − 414. i)12-s + (−498. − 181. i)13-s + (68.9 + 25.0i)14-s + (1.40e3 − 50.5i)15-s + (167. − 948. i)16-s + (189. + 327. i)17-s + ⋯
L(s)  = 1  + (−0.132 + 0.0483i)2-s + (−0.138 + 0.990i)3-s + (−0.750 + 0.629i)4-s + (−0.279 − 1.58i)5-s + (−0.0295 − 0.138i)6-s + (−0.541 − 0.454i)7-s + (0.140 − 0.242i)8-s + (−0.961 − 0.273i)9-s + (0.113 + 0.197i)10-s + (−0.273 + 1.55i)11-s + (−0.520 − 0.830i)12-s + (−0.817 − 0.297i)13-s + (0.0939 + 0.0342i)14-s + (1.61 − 0.0580i)15-s + (0.163 − 0.926i)16-s + (0.158 + 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.921 + 0.388i$
motivic weight  =  \(5\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :5/2),\ -0.921 + 0.388i)$
$L(3)$  $\approx$  $0.0148757 - 0.0735793i$
$L(\frac12)$  $\approx$  $0.0148757 - 0.0735793i$
$L(\frac{7}{2})$   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 \( 1 + (2.15 - 15.4i)T \)
good2 \( 1 + (0.752 - 0.273i)T + (24.5 - 20.5i)T^{2} \)
5 \( 1 + (15.6 + 88.6i)T + (-2.93e3 + 1.06e3i)T^{2} \)
7 \( 1 + (70.2 + 58.9i)T + (2.91e3 + 1.65e4i)T^{2} \)
11 \( 1 + (109. - 623. i)T + (-1.51e5 - 5.50e4i)T^{2} \)
13 \( 1 + (498. + 181. i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (-189. - 327. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (991. - 1.71e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-919. + 771. i)T + (1.11e6 - 6.33e6i)T^{2} \)
29 \( 1 + (1.81e3 - 662. i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (-310. + 260. i)T + (4.97e6 - 2.81e7i)T^{2} \)
37 \( 1 + (6.71e3 + 1.16e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (-7.36e3 - 2.68e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + (3.70e3 - 2.10e4i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (-3.40e3 - 2.86e3i)T + (3.98e7 + 2.25e8i)T^{2} \)
53 \( 1 - 8.28e3T + 4.18e8T^{2} \)
59 \( 1 + (7.32e3 + 4.15e4i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (-2.05e3 - 1.72e3i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (1.82e4 + 6.65e3i)T + (1.03e9 + 8.67e8i)T^{2} \)
71 \( 1 + (2.65e4 + 4.60e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (2.78e3 - 4.82e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (8.16e4 - 2.97e4i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (-2.39e4 + 8.72e3i)T + (3.01e9 - 2.53e9i)T^{2} \)
89 \( 1 + (3.51e4 - 6.08e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (1.88e4 - 1.07e5i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−16.87258560447714397970278797526, −16.13949532006340780872518262705, −14.69682072794862149737695803105, −12.85316346634617969436301461923, −12.32901656903698590387917091229, −10.06022200069098358305880546615, −9.191894503494058172977136719608, −7.86814034423758506405363929061, −5.00888847678828645590570399218, −4.06486432451144312449825592180, 0.05129383888298413077497297749, 2.84288187542617847625511604433, 5.81152209565646351220909667814, 7.05900263458330382600682207907, 8.773376761073751953874433363559, 10.51735207333025301723823272368, 11.59436546309350472171326720529, 13.34981427455726464614627533923, 14.19258192409407854562482174651, 15.31626163579999948644331618512

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