Properties

Label 2-3e3-27.4-c7-0-17
Degree $2$
Conductor $27$
Sign $-0.995 + 0.0907i$
Analytic cond. $8.43439$
Root an. cond. $2.90420$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.62 − 3.50i)2-s + (31.7 − 34.3i)3-s + (−17.6 − 14.7i)4-s + (40.7 − 230. i)5-s + (−425. + 219. i)6-s + (500. − 420. i)7-s + (773. + 1.34e3i)8-s + (−174. − 2.18e3i)9-s + (−1.20e3 + 2.08e3i)10-s + (−328. − 1.86e3i)11-s + (−1.06e3 + 136. i)12-s + (−1.36e4 + 4.95e3i)13-s + (−6.29e3 + 2.29e3i)14-s + (−6.64e3 − 8.72e3i)15-s + (−2.24e3 − 1.27e4i)16-s + (−7.31e3 + 1.26e4i)17-s + ⋯
L(s)  = 1  + (−0.851 − 0.309i)2-s + (0.678 − 0.734i)3-s + (−0.137 − 0.115i)4-s + (0.145 − 0.826i)5-s + (−0.804 + 0.415i)6-s + (0.551 − 0.462i)7-s + (0.534 + 0.925i)8-s + (−0.0798 − 0.996i)9-s + (−0.379 + 0.658i)10-s + (−0.0743 − 0.421i)11-s + (−0.178 + 0.0228i)12-s + (−1.71 + 0.625i)13-s + (−0.612 + 0.223i)14-s + (−0.508 − 0.667i)15-s + (−0.136 − 0.775i)16-s + (−0.360 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.995 + 0.0907i$
Analytic conductor: \(8.43439\)
Root analytic conductor: \(2.90420\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :7/2),\ -0.995 + 0.0907i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0433900 - 0.954579i\)
\(L(\frac12)\) \(\approx\) \(0.0433900 - 0.954579i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-31.7 + 34.3i)T \)
good2 \( 1 + (9.62 + 3.50i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (-40.7 + 230. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (-500. + 420. i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (328. + 1.86e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (1.36e4 - 4.95e3i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (7.31e3 - 1.26e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.16e4 + 3.75e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-6.19e4 - 5.19e4i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (-8.52e4 - 3.10e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (1.14e5 + 9.62e4i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (-4.02e4 + 6.96e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (-1.81e5 + 6.59e4i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (8.69e4 + 4.93e5i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (-6.15e5 + 5.16e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 + 1.14e6T + 1.17e12T^{2} \)
59 \( 1 + (-2.99e5 + 1.69e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (-1.71e6 + 1.43e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (1.17e5 - 4.27e4i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (-8.50e5 + 1.47e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (8.68e5 + 1.50e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-2.67e6 - 9.75e5i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (-2.05e6 - 7.47e5i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (5.87e6 + 1.01e7i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (-1.94e6 - 1.10e7i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.92408113280258500739233692009, −13.84752048101610556497529142583, −12.71216861250724350108842849990, −11.13280956229582325580695876924, −9.436552855913784060239254115167, −8.621535463836154252477335056271, −7.28974898611688636767580099960, −4.83574954286596179137357011947, −2.02846352674634475378219579455, −0.58835528981639288532538413865, 2.64297083877974523366848038744, 4.70254172946964401352457827369, 7.24987587903969382656664569385, 8.416160498839489867093074403395, 9.704608860814782355555457961292, 10.60597679863343949978008954116, 12.63124397023910487904320101766, 14.41096362411193846155359934178, 15.03971731963161736101800944497, 16.46674064899674352761782505620

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