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Properties

Label 2-2280-2280.1469-c0-0-1
Degree $2$
Conductor $2280$
Sign $0.356 - 0.934i$
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 + (−0.342 + 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + 0.999i·8-s + (−0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (0.642 + 0.766i)12-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.984 − 0.826i)17-s + (0.984 + 0.173i)18-s + (0.939 + 0.342i)19-s + (−0.342 + 0.939i)20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.342 + 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + 0.999i·8-s + (−0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (0.642 + 0.766i)12-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.984 − 0.826i)17-s + (0.984 + 0.173i)18-s + (0.939 + 0.342i)19-s + (−0.342 + 0.939i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 0.356 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5594110762\)
\(L(\frac12)\) \(\approx\) \(0.5594110762\)
\(L(1)\) 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 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.984 + 0.826i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.300 + 1.70i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (1.50 + 0.266i)T + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−9.304573108382036298277292691030, −8.626954745748782231830267109114, −7.903597395325695761124171059165, −7.12306744235645760941316032433, −6.39527866384974187456422957172, −5.31503560458412692321900782641, −4.82358298734764377092691701561, −3.62479648052862208774986496355, −2.77248744371716278099617990550, −0.840994869105297328409824473227, 0.837298822527157808663962189938, 1.87670867168837745676451316214, 3.15654836774042567337091871207, 3.83735389959299968076707472640, 5.21316917261079897811744817207, 6.09238518292269573647913056621, 7.20553568121515245374868377115, 7.60352099340058831011902083298, 8.086682864780731140491821048420, 9.010646330433010551317661808961

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