Properties

Label 2-2160-1.1-c3-0-91
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 24.3·7-s − 26.8·11-s − 68.2·13-s + 61.8·17-s + 75.1·19-s − 43.3·23-s + 25·25-s − 174.·29-s − 222.·31-s + 121.·35-s + 67.1·37-s + 22.2·41-s + 84.7·43-s − 585.·47-s + 251.·49-s − 38.3·53-s − 134.·55-s + 92·59-s − 226.·61-s − 341.·65-s − 858.·67-s + 116.·71-s + 911.·73-s − 655.·77-s + 285.·79-s − 999.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.31·7-s − 0.737·11-s − 1.45·13-s + 0.882·17-s + 0.907·19-s − 0.393·23-s + 0.200·25-s − 1.12·29-s − 1.29·31-s + 0.588·35-s + 0.298·37-s + 0.0845·41-s + 0.300·43-s − 1.81·47-s + 0.734·49-s − 0.0994·53-s − 0.329·55-s + 0.203·59-s − 0.474·61-s − 0.651·65-s − 1.56·67-s + 0.195·71-s + 1.46·73-s − 0.970·77-s + 0.406·79-s − 1.32·83-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: no
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 \( 1 - 5T \)
good7 \( 1 - 24.3T + 343T^{2} \)
11 \( 1 + 26.8T + 1.33e3T^{2} \)
13 \( 1 + 68.2T + 2.19e3T^{2} \)
17 \( 1 - 61.8T + 4.91e3T^{2} \)
19 \( 1 - 75.1T + 6.85e3T^{2} \)
23 \( 1 + 43.3T + 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 + 222.T + 2.97e4T^{2} \)
37 \( 1 - 67.1T + 5.06e4T^{2} \)
41 \( 1 - 22.2T + 6.89e4T^{2} \)
43 \( 1 - 84.7T + 7.95e4T^{2} \)
47 \( 1 + 585.T + 1.03e5T^{2} \)
53 \( 1 + 38.3T + 1.48e5T^{2} \)
59 \( 1 - 92T + 2.05e5T^{2} \)
61 \( 1 + 226.T + 2.26e5T^{2} \)
67 \( 1 + 858.T + 3.00e5T^{2} \)
71 \( 1 - 116.T + 3.57e5T^{2} \)
73 \( 1 - 911.T + 3.89e5T^{2} \)
79 \( 1 - 285.T + 4.93e5T^{2} \)
83 \( 1 + 999.T + 5.71e5T^{2} \)
89 \( 1 - 374.T + 7.04e5T^{2} \)
97 \( 1 + 227.T + 9.12e5T^{2} \)
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

−8.031045062360317698133057044264, −7.74344585912297469793807880142, −6.97557442620747893821136293447, −5.58566976342226000246787325249, −5.29094342185355020311232801911, −4.47182675511791707609890332874, −3.23069917940648544562306349341, −2.21502325030531071594558919439, −1.42591659551756577157957946280, 0, 1.42591659551756577157957946280, 2.21502325030531071594558919439, 3.23069917940648544562306349341, 4.47182675511791707609890332874, 5.29094342185355020311232801911, 5.58566976342226000246787325249, 6.97557442620747893821136293447, 7.74344585912297469793807880142, 8.031045062360317698133057044264

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