Properties

Label 2-2160-1.1-c3-0-66
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 4·7-s − 42·11-s + 20·13-s + 93·17-s − 59·19-s − 9·23-s + 25·25-s + 120·29-s − 47·31-s − 20·35-s − 262·37-s + 126·41-s + 178·43-s − 144·47-s − 327·49-s + 741·53-s + 210·55-s + 444·59-s + 221·61-s − 100·65-s + 538·67-s − 690·71-s − 1.12e3·73-s − 168·77-s − 665·79-s − 75·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.215·7-s − 1.15·11-s + 0.426·13-s + 1.32·17-s − 0.712·19-s − 0.0815·23-s + 1/5·25-s + 0.768·29-s − 0.272·31-s − 0.0965·35-s − 1.16·37-s + 0.479·41-s + 0.631·43-s − 0.446·47-s − 0.953·49-s + 1.92·53-s + 0.514·55-s + 0.979·59-s + 0.463·61-s − 0.190·65-s + 0.981·67-s − 1.15·71-s − 1.80·73-s − 0.248·77-s − 0.947·79-s − 0.0991·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 - 93 T + p^{3} T^{2} \)
19 \( 1 + 59 T + p^{3} T^{2} \)
23 \( 1 + 9 T + p^{3} T^{2} \)
29 \( 1 - 120 T + p^{3} T^{2} \)
31 \( 1 + 47 T + p^{3} T^{2} \)
37 \( 1 + 262 T + p^{3} T^{2} \)
41 \( 1 - 126 T + p^{3} T^{2} \)
43 \( 1 - 178 T + p^{3} T^{2} \)
47 \( 1 + 144 T + p^{3} T^{2} \)
53 \( 1 - 741 T + p^{3} T^{2} \)
59 \( 1 - 444 T + p^{3} T^{2} \)
61 \( 1 - 221 T + p^{3} T^{2} \)
67 \( 1 - 538 T + p^{3} T^{2} \)
71 \( 1 + 690 T + p^{3} T^{2} \)
73 \( 1 + 1126 T + p^{3} T^{2} \)
79 \( 1 + 665 T + p^{3} T^{2} \)
83 \( 1 + 75 T + p^{3} T^{2} \)
89 \( 1 + 1086 T + p^{3} T^{2} \)
97 \( 1 - 1544 T + p^{3} T^{2} \)
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

−8.296080780529847386824352545721, −7.64999881352461416656693596212, −6.90473632520194198134227964847, −5.82702047148670919140756963211, −5.20210041328841900767858666969, −4.24054097178934030381802617249, −3.34245607507074733993194967598, −2.42134755849513429485289203930, −1.17308091506343467959325376583, 0, 1.17308091506343467959325376583, 2.42134755849513429485289203930, 3.34245607507074733993194967598, 4.24054097178934030381802617249, 5.20210041328841900767858666969, 5.82702047148670919140756963211, 6.90473632520194198134227964847, 7.64999881352461416656693596212, 8.296080780529847386824352545721

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