Properties

Degree $2$
Conductor $25200$
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 − 6·13-s − 2·17-s − 6·19-s + 2·23-s − 6·29-s + 2·31-s − 4·37-s − 8·41-s + 4·43-s + 4·47-s + 49-s − 6·53-s + 4·59-s + 14·61-s − 4·67-s − 10·73-s + 4·77-s − 16·83-s − 8·89-s + 6·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s − 0.485·17-s − 1.37·19-s + 0.417·23-s − 1.11·29-s + 0.359·31-s − 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 1.79·61-s − 0.488·67-s − 1.17·73-s + 0.455·77-s − 1.75·83-s − 0.847·89-s + 0.628·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1590410129\)
\(L(\frac12)\) \(\approx\) \(0.1590410129\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−15.41347841393520, −14.78596173756810, −14.49312131274024, −13.60364052986888, −13.15402423065488, −12.70134094257886, −12.28990222228146, −11.56644105236899, −10.96009011831579, −10.35744981087062, −10.02445229147007, −9.359798256605521, −8.735454900271046, −8.154781489288122, −7.518814267241334, −6.968150155949892, −6.505036153500636, −5.489965266831747, −5.237385419544956, −4.433870886067094, −3.859071366132859, −2.738679460404361, −2.533978198834727, −1.640367491287382, −0.1468701738735027, 0.1468701738735027, 1.640367491287382, 2.533978198834727, 2.738679460404361, 3.859071366132859, 4.433870886067094, 5.237385419544956, 5.489965266831747, 6.505036153500636, 6.968150155949892, 7.518814267241334, 8.154781489288122, 8.735454900271046, 9.359798256605521, 10.02445229147007, 10.35744981087062, 10.96009011831579, 11.56644105236899, 12.28990222228146, 12.70134094257886, 13.15402423065488, 13.60364052986888, 14.49312131274024, 14.78596173756810, 15.41347841393520

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