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Properties

Label 2-1080-120.29-c0-0-11
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

\(L(\frac{1}{2})\) \(\approx\) \(1.634380604\)
\(L(\frac12)\) \(\approx\) \(1.634380604\)
\(L(1)\) 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

−10.00236714396569827506300111149, −9.113043645594554020425608227990, −8.397704739077227846264676819794, −7.19027279951563603464062134819, −6.26656938328989540451079772546, −5.38683451244419548524553927020, −4.57904977016929761335079120829, −3.82955732159803505065447750363, −2.50429958803416188204978141221, −1.30320287235406403392706969253, 2.39884758840921039296502451191, 3.05674886846537640048360561285, 4.30822041019378689673766774766, 5.02771584117893905065604886889, 6.25878560764349060519435445496, 6.84723679019381519342984430369, 7.38229160715719192101612220346, 8.581267364072095174355121252325, 9.267586267402594327423349960334, 10.65915443856623462155832666493

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