Properties

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

Origins

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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

\(L(1)\) \(\approx\) \(1.596055111\)
\(L(\frac12)\) \(\approx\) \(1.596055111\)
\(L(\frac{3}{2})\) 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 \( 1 + 10 T + p T^{2} \)
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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

−13.93970755217475, −13.23918417865836, −12.71910887225671, −12.16491067149236, −11.88341784731987, −11.44000228687970, −10.85082958498688, −10.04087615596593, −9.789916447217702, −9.540826952462369, −8.695193694816893, −8.332655774273482, −7.738533637423961, −7.142633510894681, −6.635535428100408, −6.266423709709159, −5.523601150043488, −5.092964641321978, −4.425829562034707, −3.642074015381062, −3.489074535328219, −2.661194235170348, −1.830187129796914, −1.380192478163516, −0.3957825969788781, 0.3957825969788781, 1.380192478163516, 1.830187129796914, 2.661194235170348, 3.489074535328219, 3.642074015381062, 4.425829562034707, 5.092964641321978, 5.523601150043488, 6.266423709709159, 6.635535428100408, 7.142633510894681, 7.738533637423961, 8.332655774273482, 8.695193694816893, 9.540826952462369, 9.789916447217702, 10.04087615596593, 10.85082958498688, 11.44000228687970, 11.88341784731987, 12.16491067149236, 12.71910887225671, 13.23918417865836, 13.93970755217475

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