Properties

Label 2-100800-1.1-c1-0-268
Degree $2$
Conductor $100800$
Sign $-1$
Analytic cond. $804.892$
Root an. cond. $28.3706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 2·11-s − 13-s + 3·17-s + 23-s − 5·29-s − 7·31-s − 2·37-s − 7·41-s + 11·43-s − 8·47-s + 49-s + 53-s + 5·59-s + 3·61-s + 12·67-s + 12·71-s + 6·73-s − 2·77-s − 10·79-s − 11·83-s + 10·89-s − 91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s + 0.208·23-s − 0.928·29-s − 1.25·31-s − 0.328·37-s − 1.09·41-s + 1.67·43-s − 1.16·47-s + 1/7·49-s + 0.137·53-s + 0.650·59-s + 0.384·61-s + 1.46·67-s + 1.42·71-s + 0.702·73-s − 0.227·77-s − 1.12·79-s − 1.20·83-s + 1.05·89-s − 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(804.892\)
Root analytic conductor: \(28.3706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.21479366605948, −13.34980230171052, −13.04605172956271, −12.57337710812733, −12.07205214136267, −11.49191326675508, −11.03951338447184, −10.64354778053726, −10.00917919381139, −9.589983073214569, −9.086426136502634, −8.420065301469041, −8.023172588438270, −7.459599576440445, −7.069145518805634, −6.439662794183338, −5.669444313691103, −5.296830854448564, −4.932880554079554, −4.029275008686960, −3.637947371884256, −2.939873451674269, −2.235332399222655, −1.707123207879919, −0.8564774153935640, 0, 0.8564774153935640, 1.707123207879919, 2.235332399222655, 2.939873451674269, 3.637947371884256, 4.029275008686960, 4.932880554079554, 5.296830854448564, 5.669444313691103, 6.439662794183338, 7.069145518805634, 7.459599576440445, 8.023172588438270, 8.420065301469041, 9.086426136502634, 9.589983073214569, 10.00917919381139, 10.64354778053726, 11.03951338447184, 11.49191326675508, 12.07205214136267, 12.57337710812733, 13.04605172956271, 13.34980230171052, 14.21479366605948

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