Properties

Degree $2$
Conductor $100800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 8·23-s + 6·29-s + 8·31-s − 2·37-s − 2·41-s + 12·43-s + 8·47-s + 49-s − 6·53-s − 4·59-s + 2·61-s − 12·67-s + 8·71-s + 14·73-s − 4·77-s + 12·83-s − 2·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s + 1.63·73-s − 0.455·77-s + 1.31·83-s − 0.211·89-s − 0.209·91-s − 1.01·97-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: $1$
Motivic weight: \(1\)
Character: $\chi_{100800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.449973859\)
\(L(\frac12)\) \(\approx\) \(2.449973859\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−13.70354659581498, −13.32758450960569, −12.67090754595710, −12.30021260066828, −11.98898089158223, −11.08118183086822, −10.80650396525614, −10.42549075796933, −9.877042531121296, −9.237890479703786, −8.792842636404467, −8.138887342990074, −7.829154842025799, −7.286725906773122, −6.653301754181981, −6.200968967807646, −5.397851355584540, −5.050891812325716, −4.550469009354584, −3.974492258567898, −2.973668099011746, −2.736164699191926, −2.102040876540121, −1.137048685449884, −0.5413319105460470, 0.5413319105460470, 1.137048685449884, 2.102040876540121, 2.736164699191926, 2.973668099011746, 3.974492258567898, 4.550469009354584, 5.050891812325716, 5.397851355584540, 6.200968967807646, 6.653301754181981, 7.286725906773122, 7.829154842025799, 8.138887342990074, 8.792842636404467, 9.237890479703786, 9.877042531121296, 10.42549075796933, 10.80650396525614, 11.08118183086822, 11.98898089158223, 12.30021260066828, 12.67090754595710, 13.32758450960569, 13.70354659581498

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