↑

Properties

Label 2-2880-20.19-c0-0-2
Degree $2$
Conductor $2880$
Sign $i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·5-s − 2i·17-s − 25-s − 49-s − 2i·53-s + 2·61-s − 2·85-s − 2·109-s − 2i·113-s + ⋯
L(s)  = 1  − i·5-s − 2i·17-s − 25-s − 49-s − 2i·53-s + 2·61-s − 2·85-s − 2·109-s − 2i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092629282\)
\(L(\frac12)\) \(\approx\) \(1.092629282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.738951663666109544115481570278, −8.126936807835944868147307998368, −7.28444923821623365285795509327, −6.56709400448014052686358093086, −5.42711424361635062957792832007, −5.01271086603588168692141118288, −4.13812732382717434903375642214, −3.09030083856249119421834944098, −2.00099519136651710013671014662, −0.69631724129609977412005697202, 1.60679091421167307449458357779, 2.61879591594576753560049850191, 3.59266650423081070027824097286, 4.24344897520284681179960376934, 5.47113422480834585219288008274, 6.20795324408342765043838458505, 6.77113518091763867275909156648, 7.69593182685508995919735106047, 8.265755569633095353080527283674, 9.138467351611885520581320754196

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