Properties

Label 2-3e3-9.2-c6-0-2
Degree $2$
Conductor $27$
Sign $0.329 - 0.944i$
Analytic cond. $6.21146$
Root an. cond. $2.49228$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.2 + 7.06i)2-s + (67.7 + 117. i)4-s + (103. − 59.9i)5-s + (−128. + 223. i)7-s + 1.00e3i·8-s + 1.69e3·10-s + (−518. − 299. i)11-s + (−968. − 1.67e3i)13-s + (−3.15e3 + 1.82e3i)14-s + (−2.79e3 + 4.84e3i)16-s − 4.87e3i·17-s + 7.68e3·19-s + (1.40e4 + 8.12e3i)20-s + (−4.22e3 − 7.32e3i)22-s + (594. − 343. i)23-s + ⋯
L(s)  = 1  + (1.52 + 0.882i)2-s + (1.05 + 1.83i)4-s + (0.830 − 0.479i)5-s + (−0.375 + 0.650i)7-s + 1.97i·8-s + 1.69·10-s + (−0.389 − 0.224i)11-s + (−0.440 − 0.763i)13-s + (−1.14 + 0.663i)14-s + (−0.682 + 1.18i)16-s − 0.992i·17-s + 1.12·19-s + (1.75 + 1.01i)20-s + (−0.397 − 0.687i)22-s + (0.0488 − 0.0282i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(6.21146\)
Root analytic conductor: \(2.49228\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.97978 + 2.11523i\)
\(L(\frac12)\) \(\approx\) \(2.97978 + 2.11523i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-12.2 - 7.06i)T + (32 + 55.4i)T^{2} \)
5 \( 1 + (-103. + 59.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (128. - 223. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (518. + 299. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (968. + 1.67e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 4.87e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.68e3T + 4.70e7T^{2} \)
23 \( 1 + (-594. + 343. i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.99e4 + 1.73e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.38e3 + 2.40e3i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 1.71e4T + 2.56e9T^{2} \)
41 \( 1 + (7.15e4 - 4.13e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (6.32e4 - 1.09e5i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-7.20e4 - 4.15e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 4.00e4iT - 2.21e10T^{2} \)
59 \( 1 + (-5.76e4 + 3.32e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.84e5 - 3.19e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.27e5 + 2.21e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 2.69e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.51e5T + 1.51e11T^{2} \)
79 \( 1 + (-3.55e5 + 6.14e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (3.01e5 + 1.74e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + 8.36e5iT - 4.96e11T^{2} \)
97 \( 1 + (4.70e4 - 8.14e4i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−15.96586322102358318666612622935, −14.99464795670260556896308797850, −13.67242647787594667684135245583, −12.98384112448284244301202818961, −11.78612428192529888161875800963, −9.520039371731718702757019757691, −7.61023053184990458951289261054, −5.92751989994942189505152341744, −5.08471949449368559494640849524, −2.92003064202040674685120633271, 1.98235432910698225670101650146, 3.65702259340915853468927871094, 5.38092182516226455454686350498, 6.83000615608205668523734305530, 9.841408692747138538782020713016, 10.79283520160082938376379525523, 12.19523753356736156012501509200, 13.40240872460601434144562715745, 14.08795653766865631676617574139, 15.21745865237627371767158767527

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