Properties

Label 2-1080-120.29-c0-0-8
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $0.538990$
Root an. cond. $0.734159$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s + 13-s + 16-s − 17-s − 20-s + 22-s − 23-s + 25-s + 26-s + 29-s − 31-s + 32-s − 34-s − 2·37-s − 40-s + 43-s + 44-s − 46-s − 47-s + 49-s + 50-s + 52-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s + 13-s + 16-s − 17-s − 20-s + 22-s − 23-s + 25-s + 26-s + 29-s − 31-s + 32-s − 34-s − 2·37-s − 40-s + 43-s + 44-s − 46-s − 47-s + 49-s + 50-s + 52-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1080} (269, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.743834889\)
\(L(\frac12)\) \(\approx\) \(1.743834889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.50638804180882110024019075215, −9.076059257102549727431218774257, −8.364744468505862996650656791873, −7.36016744616500690164238977722, −6.63414111117019868014339546332, −5.86560143986269548075107847744, −4.60110559235323879549108954537, −3.97710216374685585709799224813, −3.17507501705780755725605470747, −1.65451306963902888625227182249, 1.65451306963902888625227182249, 3.17507501705780755725605470747, 3.97710216374685585709799224813, 4.60110559235323879549108954537, 5.86560143986269548075107847744, 6.63414111117019868014339546332, 7.36016744616500690164238977722, 8.364744468505862996650656791873, 9.076059257102549727431218774257, 10.50638804180882110024019075215

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