Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 4·11-s + 6·13-s + 2·17-s + 4·19-s + 8·23-s − 2·29-s − 10·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s + 4·59-s − 6·61-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s + 4·83-s + 6·89-s + 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.371·29-s − 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 0.768·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s + 0.439·83-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 + 0.0940·113-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.79590721327510, −13.53844178598431, −13.06587184662273, −12.66393347878448, −11.97210910416053, −11.48281923025532, −11.01483128379171, −10.57493916290835, −10.26639965015180, −9.440808876865963, −8.986686862886365, −8.561761793464640, −7.938437376439034, −7.581134148700782, −6.992483499591601, −6.403113692087578, −5.714473378943728, −5.310506123124622, −4.901493940263330, −4.137655131590052, −3.344334315361931, −3.154465338091315, −2.329261087515958, −1.427322021954311, −1.063035661429207, 0, 1.063035661429207, 1.427322021954311, 2.329261087515958, 3.154465338091315, 3.344334315361931, 4.137655131590052, 4.901493940263330, 5.310506123124622, 5.714473378943728, 6.403113692087578, 6.992483499591601, 7.581134148700782, 7.938437376439034, 8.561761793464640, 8.986686862886365, 9.440808876865963, 10.26639965015180, 10.57493916290835, 11.01483128379171, 11.48281923025532, 11.97210910416053, 12.66393347878448, 13.06587184662273, 13.53844178598431, 13.79590721327510

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