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Properties

Label 2-3e3-9.5-c8-0-5
Degree $2$
Conductor $27$
Sign $0.981 + 0.189i$
Analytic cond. $10.9992$
Root an. cond. $3.31650$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.0 − 9.86i)2-s + (66.7 − 115. i)4-s + (896. + 517. i)5-s + (−21.9 − 37.9i)7-s + 2.41e3i·8-s + 2.04e4·10-s + (1.52e4 − 8.81e3i)11-s + (616. − 1.06e3i)13-s + (−749. − 432. i)14-s + (4.09e4 + 7.09e4i)16-s − 1.06e5i·17-s − 1.15e5·19-s + (1.19e5 − 6.91e4i)20-s + (1.73e5 − 3.01e5i)22-s + (9.76e4 + 5.63e4i)23-s + ⋯
L(s)  = 1  + (1.06 − 0.616i)2-s + (0.260 − 0.451i)4-s + (1.43 + 0.828i)5-s + (−0.00912 − 0.0158i)7-s + 0.590i·8-s + 2.04·10-s + (1.04 − 0.601i)11-s + (0.0215 − 0.0373i)13-s + (−0.0194 − 0.0112i)14-s + (0.624 + 1.08i)16-s − 1.27i·17-s − 0.888·19-s + (0.748 − 0.432i)20-s + (0.742 − 1.28i)22-s + (0.348 + 0.201i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(10.9992\)
Root analytic conductor: \(3.31650\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :4),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.65716 - 0.350487i\)
\(L(\frac12)\) \(\approx\) \(3.65716 - 0.350487i\)
\(L(5)\) 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 \( 1 + (-17.0 + 9.86i)T + (128 - 221. i)T^{2} \)
5 \( 1 + (-896. - 517. i)T + (1.95e5 + 3.38e5i)T^{2} \)
7 \( 1 + (21.9 + 37.9i)T + (-2.88e6 + 4.99e6i)T^{2} \)
11 \( 1 + (-1.52e4 + 8.81e3i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + (-616. + 1.06e3i)T + (-4.07e8 - 7.06e8i)T^{2} \)
17 \( 1 + 1.06e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.15e5T + 1.69e10T^{2} \)
23 \( 1 + (-9.76e4 - 5.63e4i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + (7.20e5 - 4.16e5i)T + (2.50e11 - 4.33e11i)T^{2} \)
31 \( 1 + (-1.28e5 + 2.22e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + 2.84e6T + 3.51e12T^{2} \)
41 \( 1 + (2.51e6 + 1.45e6i)T + (3.99e12 + 6.91e12i)T^{2} \)
43 \( 1 + (-7.91e5 - 1.37e6i)T + (-5.84e12 + 1.01e13i)T^{2} \)
47 \( 1 + (1.36e6 - 7.89e5i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + 8.77e6iT - 6.22e13T^{2} \)
59 \( 1 + (-8.09e6 - 4.67e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (2.86e6 + 4.95e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-3.95e6 + 6.84e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 3.25e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.75e7T + 8.06e14T^{2} \)
79 \( 1 + (-6.16e5 - 1.06e6i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (-4.84e7 + 2.79e7i)T + (1.12e15 - 1.95e15i)T^{2} \)
89 \( 1 - 4.18e7iT - 3.93e15T^{2} \)
97 \( 1 + (-2.48e7 - 4.29e7i)T + (-3.91e15 + 6.78e15i)T^{2} \)
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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

−14.79698040553658342590611657770, −13.99016175779117897755691399182, −13.23453702535625451204219227070, −11.71732190331489193378778998488, −10.58647616043940950334593769914, −9.092013815991154745754100391723, −6.64138198409455983747183685594, −5.33024257546953507968451565022, −3.38476162660253113745477872581, −1.98105653337556451936822471922, 1.60119755252866995424809139168, 4.23311092341880382570261662914, 5.59935125037535467296268155453, 6.62019607804354122495408648593, 8.926021507661952358395568133991, 10.13688778963327032883474911499, 12.40081227382164274580167738986, 13.20161152550644203256861383410, 14.23922467644256165680286925711, 15.21507697556089821742118063923

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