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Properties

Label 2-1080-40.13-c0-0-3
Degree $2$
Conductor $1080$
Sign $-0.287 + 0.957i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + (1.36 − 1.36i)7-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + 0.517i·11-s − 1.93·14-s − 1.00·16-s + (0.965 − 0.258i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (1.36 + 1.36i)28-s + 1.41·29-s − 1.73·31-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + (1.36 − 1.36i)7-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + 0.517i·11-s − 1.93·14-s − 1.00·16-s + (0.965 − 0.258i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (1.36 + 1.36i)28-s + 1.41·29-s − 1.73·31-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ -0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7844751762\)
\(L(\frac12)\) \(\approx\) \(0.7844751762\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.258 + 0.965i)T \)
good7 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + 1.73T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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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

−9.928626727884205576597442055766, −8.980103115644693778664470101935, −8.226872879213923295687644626465, −7.64375473591729022761042688060, −6.92376047343167741283187624648, −5.14438721890602614411001673211, −4.42662242455967796132914608374, −3.70219815854736058990619275005, −1.96519703603731467808733284667, −1.01529383903973130486575573420, 1.77020091159882775649744440621, 2.83273214162023379938068175548, 4.47882876538452203911135708918, 5.50702792867651021988004776386, 6.08958346000417063334713675968, 7.15719551827786237150775450688, 7.909502702460712127207740414830, 8.588699213813034586098668240221, 9.220861584802162907005388847447, 10.35801778360859512956876749976

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