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Properties

Label 2-1140-1140.599-c0-0-2
Degree $2$
Conductor $1140$
Sign $0.624 - 0.780i$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
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 + (0.766 − 0.642i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 0.342i)5-s + (0.939 + 0.342i)6-s − 0.999·8-s + (0.173 − 0.984i)9-s + (0.173 + 0.984i)10-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.0603 + 0.342i)17-s + (0.939 − 0.342i)18-s + (−0.766 + 0.642i)19-s + (−0.766 + 0.642i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 0.342i)5-s + (0.939 + 0.342i)6-s − 0.999·8-s + (0.173 − 0.984i)9-s + (0.173 + 0.984i)10-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.0603 + 0.342i)17-s + (0.939 − 0.342i)18-s + (−0.766 + 0.642i)19-s + (−0.766 + 0.642i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781053162\)
\(L(\frac12)\) \(\approx\) \(1.781053162\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T^{2} \)
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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

−9.914005511268239759635328242824, −8.987848510360713150765906674712, −8.406267326433586941131303521904, −7.59255585974611883391054223420, −6.52631126321286132891504309192, −6.33118914321715665015185843284, −5.16015635565726985241479745903, −3.97742174706645722329848264129, −2.97227435955386582859502116133, −1.92479698122042120071435656208, 1.72542665164866642867341310879, 2.57603833108133099118199150600, 3.64447222543108844759740113786, 4.57018570007465356266668548596, 5.36727245534912805075663488030, 6.18648461968303232443023765166, 7.57020523099813467370636162951, 8.680639781681495345485287520178, 9.395518223984065197378123953501, 9.751368789804890303303097482048

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