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Properties

Label 2-1080-120.29-c0-0-0
Degree $2$
Conductor $1080$
Sign $-0.866 - 0.5i$
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  + i·2-s − 4-s + (0.866 + 0.5i)5-s + 1.73i·7-s − i·8-s + (−0.5 + 0.866i)10-s − 1.73·11-s − 1.73·14-s + 16-s + (−0.866 − 0.5i)20-s − 1.73i·22-s + (0.499 + 0.866i)25-s − 1.73i·28-s + 31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.866 + 0.5i)5-s + 1.73i·7-s − i·8-s + (−0.5 + 0.866i)10-s − 1.73·11-s − 1.73·14-s + 16-s + (−0.866 − 0.5i)20-s − 1.73i·22-s + (0.499 + 0.866i)25-s − 1.73i·28-s + 31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.866 - 0.5i$
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: \((2,\ 1080,\ (\ :0),\ -0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9271061268\)
\(L(\frac12)\) \(\approx\) \(0.9271061268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.08322887491663591233966540071, −9.543364995404761822417107962717, −8.597242644480923531800374545078, −8.054150553360396994732463364170, −6.97021515484389462163631641657, −6.05106501151413431306618060982, −5.49588552536215980422463874785, −4.86994752018897555580602482898, −3.09842598395990234858796801035, −2.24268083273325330541403018118, 0.869459263362982107627160513250, 2.20834013717454747780806594918, 3.32062298491310001628278791586, 4.51904233837962069829048957274, 5.06209797470060113091615043386, 6.19186171240020255060742843792, 7.51778571151360962519223862736, 8.132031209679243081392231242504, 9.180057933832893529515346655209, 10.10054753173865492609566976555

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