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Properties

Label 2-1140-1140.1139-c0-0-4
Degree $2$
Conductor $1140$
Sign $0.382 - 0.923i$
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  + (0.382 + 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 + 0.707i)4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.923 − 0.382i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + 1.41·11-s + (0.382 − 0.923i)12-s + 0.765·13-s + (0.382 + 0.923i)15-s − i·16-s + (0.923 + 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 + 0.707i)4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.923 − 0.382i)8-s + (0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + 1.41·11-s + (0.382 − 0.923i)12-s + 0.765·13-s + (0.382 + 0.923i)15-s − i·16-s + (0.923 + 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9551614118\)
\(L(\frac12)\) \(\approx\) \(0.9551614118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + iT \)
19 \( 1 - iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - 0.765T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.84T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.765iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + 1.84iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.84T + T^{2} \)
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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

−9.840854800993669846732447154560, −9.304107588033336664350877536090, −8.511380339659927993763083546076, −7.62606413318184490869226296227, −6.42230681579161074916240323831, −6.05330533251037075732837766213, −5.11995561988910963390156975545, −4.23133568237963473973401604550, −3.69583485419788335373336351084, −1.20530155597468515230020167679, 1.20332223702573974594037051862, 2.45818643447057204265874486048, 3.68925348338802590374948159412, 4.48841623769361823810937163536, 5.68734204526909340310778751654, 6.40476291097672002054643777098, 6.99894164116131941288313583007, 8.299317278703548183785096284205, 9.409071946241769926476722638979, 10.05388121457207341137597500430

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