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Properties

Label 2-2280-2280.1019-c0-0-1
Degree $2$
Conductor $2280$
Sign $0.582 - 0.813i$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.499i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.582 - 0.813i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 0.582 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.321906306\)
\(L(\frac12)\) \(\approx\) \(2.321906306\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.877042733969384921656465505016, −8.370875849070038573106877587971, −7.59933552927184822090306189603, −7.06302164851410156027885233510, −6.43207976545182783963571963334, −5.38184734607222233151067264759, −4.31752805424807140184445634346, −3.48856923427398593848118057442, −3.03275513006769400902655131527, −1.76413024488460118182245408806, 1.27452106443419194414054088915, 2.69366926586840131284632843457, 3.31543636472585341741301699676, 4.19923789622440569034677484028, 4.84861184439535711802950320459, 5.52938692528997917763054119643, 6.91407860474367271981769783210, 7.49525858531761550599893381329, 8.411400507298151637257466083938, 9.187766015056056404791682491396

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