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Properties

Label 2-3e3-9.7-c3-0-1
Degree $2$
Conductor $27$
Sign $0.118 + 0.993i$
Analytic cond. $1.59305$
Root an. cond. $1.26216$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.18 − 3.78i)2-s + (−5.55 − 9.62i)4-s + (2.31 + 4.00i)5-s + (−6.05 + 10.4i)7-s − 13.6·8-s + 20.2·10-s + (5.01 − 8.67i)11-s + (24.2 + 42.0i)13-s + (26.4 + 45.8i)14-s + (14.6 − 25.4i)16-s − 75.3·17-s − 116.·19-s + (25.7 − 44.5i)20-s + (−21.9 − 37.9i)22-s + (−19.0 − 32.9i)23-s + ⋯
L(s)  = 1  + (0.772 − 1.33i)2-s + (−0.694 − 1.20i)4-s + (0.206 + 0.358i)5-s + (−0.327 + 0.566i)7-s − 0.602·8-s + 0.639·10-s + (0.137 − 0.237i)11-s + (0.518 + 0.897i)13-s + (0.505 + 0.875i)14-s + (0.229 − 0.397i)16-s − 1.07·17-s − 1.40·19-s + (0.287 − 0.498i)20-s + (−0.212 − 0.367i)22-s + (−0.172 − 0.298i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.118 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.118 + 0.993i$
Analytic conductor: \(1.59305\)
Root analytic conductor: \(1.26216\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3/2),\ 0.118 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.21726 - 1.08114i\)
\(L(\frac12)\) \(\approx\) \(1.21726 - 1.08114i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-2.18 + 3.78i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-2.31 - 4.00i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (6.05 - 10.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-5.01 + 8.67i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 - 42.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 75.3T + 4.91e3T^{2} \)
19 \( 1 + 116.T + 6.85e3T^{2} \)
23 \( 1 + (19.0 + 32.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (11.3 - 19.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (15.0 + 26.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 130.T + 5.06e4T^{2} \)
41 \( 1 + (-173. - 300. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (13.3 - 23.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-230. + 399. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 438.T + 1.48e5T^{2} \)
59 \( 1 + (-4.18 - 7.24i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 + 71.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (341. + 591. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.09e3T + 3.57e5T^{2} \)
73 \( 1 - 470.T + 3.89e5T^{2} \)
79 \( 1 + (243. - 420. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-49.5 + 85.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 8.80T + 7.04e5T^{2} \)
97 \( 1 + (330. - 572. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−16.52028607045628524172164355630, −14.92787385430361298138781226323, −13.73552702169019275909827147915, −12.71257491934974558488827800823, −11.50727943786206024210616596177, −10.49391774784210497496531479052, −8.944986652991427055255157976930, −6.32298392977254693961268418088, −4.25871935047567256819406160559, −2.37754788480497373514455004333, 4.23142484728754545194101408817, 5.87130882116072105145105332121, 7.19270388659881643232665188608, 8.703722026946425689068046230621, 10.65333115366953732278445694551, 12.81677667740808967089611965998, 13.48513780986124751144015883020, 14.84705613719766784111740531515, 15.77530239802932358428905697381, 16.88874695126329030137549750624

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