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Properties

Label 2-1140-1140.659-c0-0-0
Degree $2$
Conductor $1140$
Sign $0.500 - 0.865i$
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  + (−0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (0.5 + 0.866i)6-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)10-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−1.17 + 0.984i)17-s + (−0.173 − 0.984i)18-s + (−0.939 − 0.342i)19-s + (−0.939 + 0.342i)20-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (0.5 + 0.866i)6-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)10-s + (0.173 − 0.984i)12-s + (0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−1.17 + 0.984i)17-s + (−0.173 − 0.984i)18-s + (−0.939 − 0.342i)19-s + (−0.939 + 0.342i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.500 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4242170665\)
\(L(\frac12)\) \(\approx\) \(0.4242170665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (1.26 - 1.50i)T + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 - 0.984i)T^{2} \)
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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

−10.55524240135337861588668593725, −9.449257420007855639110340093348, −8.634468263296852841393050898635, −7.57639454010119850018838422949, −6.79949119822350186545612150775, −6.35607548831298956138333526673, −4.97267648814143284360314875615, −3.85032332163138060793032143661, −2.63633774048080007628182801784, −1.56635992517233843912082678809, 0.55880017182110823209056818561, 2.05153202996447081638289548372, 4.17692022526029091512145866286, 4.99129960222279770071393545963, 5.61670220208197133724409856344, 6.59684174262021862961718375568, 7.23488675188690708696073769297, 8.446148676259495923741684303477, 9.081516826488705114533794476821, 9.694834925818535849971282618810

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