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Properties

Label 2-3e3-9.2-c4-0-0
Degree $2$
Conductor $27$
Sign $-0.577 - 0.816i$
Analytic cond. $2.79098$
Root an. cond. $1.67062$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.86 − 2.23i)2-s + (1.96 + 3.41i)4-s + (−13.8 + 8.01i)5-s + (−36.2 + 62.7i)7-s + 53.8i·8-s + 71.5·10-s + (−83.2 − 48.0i)11-s + (−76.9 − 133. i)13-s + (280. − 161. i)14-s + (151. − 262. i)16-s + 72.7i·17-s − 190.·19-s + (−54.6 − 31.5i)20-s + (214. + 371. i)22-s + (12.5 − 7.22i)23-s + ⋯
L(s)  = 1  + (−0.966 − 0.558i)2-s + (0.123 + 0.213i)4-s + (−0.555 + 0.320i)5-s + (−0.739 + 1.28i)7-s + 0.841i·8-s + 0.715·10-s + (−0.688 − 0.397i)11-s + (−0.455 − 0.788i)13-s + (1.43 − 0.825i)14-s + (0.592 − 1.02i)16-s + 0.251i·17-s − 0.528·19-s + (−0.136 − 0.0788i)20-s + (0.443 + 0.768i)22-s + (0.0236 − 0.0136i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(2.79098\)
Root analytic conductor: \(1.67062\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :2),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0786064 + 0.151803i\)
\(L(\frac12)\) \(\approx\) \(0.0786064 + 0.151803i\)
\(L(3)\) 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 \( 1 + (3.86 + 2.23i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + (13.8 - 8.01i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (36.2 - 62.7i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (83.2 + 48.0i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (76.9 + 133. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 72.7iT - 8.35e4T^{2} \)
19 \( 1 + 190.T + 1.30e5T^{2} \)
23 \( 1 + (-12.5 + 7.22i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (620. + 358. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-151. - 262. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 826.T + 1.87e6T^{2} \)
41 \( 1 + (481. - 278. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (446. - 773. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-3.42e3 - 1.97e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.96e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.68e3 - 2.70e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-856. + 1.48e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.31e3 - 4.01e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 6.69e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.82e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.86e3 + 4.96e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (2.45e3 + 1.41e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.42e4iT - 6.27e7T^{2} \)
97 \( 1 + (3.58e3 - 6.20e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−17.31222029345172251823905923387, −15.77946693252305306359042710678, −14.84751253428308425183926921327, −12.91395003812320189314993778792, −11.63398923612921083626909152361, −10.40749820478836468046449260872, −9.159710865437375978384400152462, −7.925401650074706677908405369262, −5.68309964657096455190355330759, −2.72951115890856146166344543448, 0.17568638437198974052960598967, 4.11415363409533878871646727571, 6.86688861635046238011152429020, 7.82954227075206347430519734291, 9.363768276562894302235571884812, 10.50606015285606503417995404710, 12.41259396553642377565216072913, 13.60997802394700123790774857884, 15.41247650620529943868861701413, 16.49249001152967374814739990424

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