Properties

Label 2-3e3-27.11-c2-0-0
Degree $2$
Conductor $27$
Sign $-0.983 - 0.181i$
Analytic cond. $0.735696$
Root an. cond. $0.857727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 + 2.59i)2-s + (−2.99 + 0.0504i)3-s + (−1.30 − 7.38i)4-s + (−1.19 + 3.28i)5-s + (6.40 − 7.90i)6-s + (−1.88 + 10.7i)7-s + (10.2 + 5.92i)8-s + (8.99 − 0.302i)9-s + (−5.92 − 10.2i)10-s + (0.00845 + 0.0232i)11-s + (4.27 + 22.0i)12-s + (6.61 − 5.55i)13-s + (−23.6 − 28.2i)14-s + (3.42 − 9.91i)15-s + (−9.59 + 3.49i)16-s + (−3.55 + 2.05i)17-s + ⋯
L(s)  = 1  + (−1.08 + 1.29i)2-s + (−0.999 + 0.0168i)3-s + (−0.325 − 1.84i)4-s + (−0.239 + 0.656i)5-s + (1.06 − 1.31i)6-s + (−0.269 + 1.52i)7-s + (1.28 + 0.740i)8-s + (0.999 − 0.0336i)9-s + (−0.592 − 1.02i)10-s + (0.000768 + 0.00211i)11-s + (0.356 + 1.83i)12-s + (0.508 − 0.427i)13-s + (−1.69 − 2.01i)14-s + (0.228 − 0.660i)15-s + (−0.599 + 0.218i)16-s + (−0.209 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(0.735696\)
Root analytic conductor: \(0.857727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :1),\ -0.983 - 0.181i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0325082 + 0.355662i\)
\(L(\frac12)\) \(\approx\) \(0.0325082 + 0.355662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 - 0.0504i)T \)
good2 \( 1 + (2.17 - 2.59i)T + (-0.694 - 3.93i)T^{2} \)
5 \( 1 + (1.19 - 3.28i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (1.88 - 10.7i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (-0.00845 - 0.0232i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (-6.61 + 5.55i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (3.55 - 2.05i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (12.2 - 21.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.41 + 1.30i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (-12.4 + 14.8i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (1.89 + 10.7i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (5.30 + 9.19i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-49.7 - 59.2i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (18.6 - 6.77i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (-45.8 - 8.08i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 + 39.1iT - 2.80e3T^{2} \)
59 \( 1 + (-17.6 + 48.4i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-1.84 + 10.4i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-81.7 + 68.5i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (42.1 - 24.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (8.15 - 14.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-55.8 - 46.8i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (59.6 - 71.0i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (-38.0 - 21.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (88.2 - 32.1i)T + (7.20e3 - 6.04e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.70434204043162146036904526361, −16.49386394918226439368257514829, −15.56510875853222601652688926253, −14.84650448541235147511595830809, −12.53887845487082187526571101435, −11.00819710410905577837251025959, −9.647678334913479480617839556938, −8.189162651482107326984126588893, −6.57036893846826254802442919311, −5.67172418859031670886391691590, 0.795455365553394133549567097715, 4.24291892994181507673228199030, 7.10536545284400575231454497058, 8.914657444566343362921122010163, 10.38859749163581936493351361018, 11.07697282226890428462893125871, 12.35358470944991135596415183120, 13.39278063648640943386378879749, 16.10634009170172795772004522745, 17.00611080804080191961084896164

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