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Properties

Label 2-1140-1140.599-c0-0-1
Degree $2$
Conductor $1140$
Sign $0.877 + 0.479i$
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  + (0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)10-s + (0.939 − 0.342i)12-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)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  + (0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 + 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)10-s + (0.939 − 0.342i)12-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.877 + 0.479i$
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: \((2,\ 1140,\ (\ :0),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9615661287\)
\(L(\frac12)\) \(\approx\) \(0.9615661287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
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

−9.858784877470565800169290773406, −9.629049191890658364464551695042, −8.794424827821127425767521870971, −7.38729315479922353028128403688, −6.19345408328705223284087851566, −5.53875115391409935585072126186, −4.79385349856933467840278776744, −3.69817058400049785024243726740, −2.74125516439776482234675452737, −1.31780697226353975687295496959, 1.16699277088853932588220560372, 2.80190478803123132161729357861, 4.49728977762319956997039241552, 5.18410177280954922253806970588, 5.94661993496502059997627947558, 6.62223536710575564081266584381, 7.36001627652867132473437512815, 8.335561874372114107713630150447, 9.047870414931447407285208260350, 10.05572597589645099431725524888

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