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Properties

Label 2-1140-1140.299-c0-0-2
Degree $2$
Conductor $1140$
Sign $0.992 - 0.120i$
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  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−1.76 + 0.642i)17-s + (−0.766 − 0.642i)18-s + (0.173 − 0.984i)19-s + (0.173 + 0.984i)20-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−1.76 + 0.642i)17-s + (−0.766 − 0.642i)18-s + (0.173 − 0.984i)19-s + (0.173 + 0.984i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.992 - 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.016700192\)
\(L(\frac12)\) \(\approx\) \(2.016700192\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T^{2} \)
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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

−10.22692169744154521451119349581, −8.875710632301924478486151126693, −8.284841080477857666040780376397, −7.10666305874296059835190546831, −6.64773779627727530051088504511, −6.06271106929223900255117572680, −5.03414320582679136947087876093, −3.83765015120258369885762574274, −2.56606432007017307272747490266, −2.06271561657271292546252327537, 1.83185543955267751264850925589, 2.85813095234502812900370461881, 4.09282849065469521696614293210, 4.63492407228788408949253171140, 5.65627894559975545300420874530, 6.11224447929876240398484430799, 7.44957530241000569719861017804, 8.609575028670677120204245603641, 9.486887483525380106798003778426, 9.942021688016363899761212577482

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