Properties

Label 2-1080-120.29-c0-0-1
Degree $2$
Conductor $1080$
Sign $-i$
Analytic cond. $0.538990$
Root an. cond. $0.734159$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73·17-s + 1.73i·19-s + (0.866 + 0.499i)20-s − 1.73·23-s − 25-s + 31-s + (0.866 + 0.499i)32-s + (−1.49 + 0.866i)34-s + (−0.866 − 1.49i)38-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73·17-s + 1.73i·19-s + (0.866 + 0.499i)20-s − 1.73·23-s − 25-s + 31-s + (0.866 + 0.499i)32-s + (−1.49 + 0.866i)34-s + (−0.866 − 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-i$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6819262146\)
\(L(\frac12)\) \(\approx\) \(0.6819262146\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.10405513083396779112910541810, −9.748336323406772493099518654573, −8.397692781226793200637204925531, −7.81188221421039848535023044515, −7.16960425272490589735042832760, −5.95125536030593633412552129607, −5.78487945793853369233458768088, −4.06366637854581671650160072314, −2.88950496674166066815577695259, −1.60710571489152586030230424077, 0.893866419902744668565386984301, 2.21923413693686383482525452235, 3.47105853037303369137181192217, 4.51567241286491564142400744700, 5.59601600180493254397855074635, 6.69218734271625664329966040459, 7.78951021353045043087839930429, 8.242405793562806391399728382769, 9.171085672457500616098977651728, 9.779681599709291891197608617088

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