Properties

Label 2-1080-40.19-c0-0-2
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)20-s − 23-s + 25-s − 1.73i·31-s + (0.499 − 0.866i)32-s + (−1.49 + 0.866i)34-s + (0.5 + 0.866i)38-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)20-s − 23-s + 25-s − 1.73i·31-s + (0.499 − 0.866i)32-s + (−1.49 + 0.866i)34-s + (0.5 + 0.866i)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: $\chi_{1080} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434449014\)
\(L(\frac12)\) \(\approx\) \(1.434449014\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.967094511630002132362033973609, −9.501704922783443986962260404163, −8.397507380657265063768272422563, −7.84204171831906188317733810386, −6.67597027772975196912052189636, −6.03987122400830339625076356985, −5.38717823918188904608958474212, −4.30448062350206923087347667575, −3.29848818436494529445248891710, −1.93598480651231218226739935785, 1.34225732997095697318256003625, 2.54832594156346273710871363919, 3.37979763031487280795803323210, 4.80442440001438324094990063770, 5.31259196135914591160031334551, 6.28957581810395958815907928466, 7.19244977660992586847552213955, 8.542128382364495245659577516935, 9.416313602098014119626519050808, 9.886282222828552205782785037272

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