Properties

Label 2-100800-1.1-c1-0-200
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 − 4·11-s − 6·13-s + 6·17-s + 4·23-s − 6·29-s + 2·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s + 6·53-s − 12·59-s + 10·61-s − 4·67-s + 8·71-s + 14·73-s + 4·77-s − 8·79-s − 12·83-s + 6·89-s + 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.834·23-s − 1.11·29-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 1.63·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-s + 0.635·89-s + 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−13.98871777026993, −13.45960520873383, −12.79607255272711, −12.72367749359711, −12.09135662458391, −11.61846174706405, −11.03046654723440, −10.42799775015072, −10.02886065119166, −9.646390913500967, −9.191528611892488, −8.441415356659871, −7.817193348099529, −7.582629235693824, −7.036955725825645, −6.485981641803324, −5.633803671885383, −5.294483800413331, −4.939831715523025, −4.177530888411601, −3.392328538664539, −2.955612703990040, −2.398491820105141, −1.705837281456300, −0.7236360151609874, 0, 0.7236360151609874, 1.705837281456300, 2.398491820105141, 2.955612703990040, 3.392328538664539, 4.177530888411601, 4.939831715523025, 5.294483800413331, 5.633803671885383, 6.485981641803324, 7.036955725825645, 7.582629235693824, 7.817193348099529, 8.441415356659871, 9.191528611892488, 9.646390913500967, 10.02886065119166, 10.42799775015072, 11.03046654723440, 11.61846174706405, 12.09135662458391, 12.72367749359711, 12.79607255272711, 13.45960520873383, 13.98871777026993

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