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Properties

Label 2-2280-2280.2189-c0-0-3
Degree $2$
Conductor $2280$
Sign $-0.790 + 0.612i$
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 + (−0.984 + 0.173i)3-s + (0.499 − 0.866i)4-s + (−0.642 − 0.766i)5-s + (−0.766 + 0.642i)6-s − 0.999i·8-s + (0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (−0.342 + 0.939i)12-s + (0.766 + 0.642i)15-s + (−0.5 − 0.866i)16-s + (0.642 + 0.233i)17-s + (0.642 − 0.766i)18-s + (−0.173 − 0.984i)19-s + (−0.984 + 0.173i)20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.984 + 0.173i)3-s + (0.499 − 0.866i)4-s + (−0.642 − 0.766i)5-s + (−0.766 + 0.642i)6-s − 0.999i·8-s + (0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (−0.342 + 0.939i)12-s + (0.766 + 0.642i)15-s + (−0.5 − 0.866i)16-s + (0.642 + 0.233i)17-s + (0.642 − 0.766i)18-s + (−0.173 − 0.984i)19-s + (−0.984 + 0.173i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (2189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078154741\)
\(L(\frac12)\) \(\approx\) \(1.078154741\)
\(L(1)\) 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 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 + (0.642 + 0.766i)T \)
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 \( 1 + (-0.642 - 0.233i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (1.32 + 1.11i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-1.20 + 1.43i)T + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
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 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
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

−9.061530803829175098386023282774, −8.061095725242309848944741077075, −7.10646567230092054105757267757, −6.31135785830731665206854258584, −5.56869046659898620560338306140, −4.75881801760645573639574035556, −4.26655724953542284492924580604, −3.36419466241226147551935355289, −1.91810932016313597134758444884, −0.62465505713173181634435956989, 1.84031266201673028909514047072, 3.21135560212293272170769957832, 3.95355700712657184487834149471, 4.78348030702557850581060254528, 5.79636478587272272853724518631, 6.18933945174162523124461593334, 7.12466427179563983963646733313, 7.66567356742259493092075511798, 8.262752524909803084047733849204, 9.720967453934272745612624129420

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