Properties

Degree $2$
Conductor $25200$
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 + 2·13-s − 6·17-s + 8·23-s − 10·29-s + 8·31-s − 2·37-s + 2·41-s + 8·43-s − 4·47-s + 49-s + 10·53-s + 4·59-s − 6·61-s − 12·71-s + 6·73-s + 4·77-s + 8·79-s + 4·83-s − 14·89-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.66·23-s − 1.85·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.520·59-s − 0.768·61-s − 1.42·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.439·83-s − 1.48·89-s − 0.209·91-s − 0.203·97-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: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

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

−15.45459671355163, −15.27000594480363, −14.73811365764453, −13.76149942117291, −13.43843896836280, −13.02234060774735, −12.61195127029303, −11.77244404368700, −11.18340640183898, −10.74191564527821, −10.37693478545647, −9.378800090854644, −9.198843671572595, −8.419007952844940, −7.950512348243569, −7.056496054815601, −6.880464833966881, −5.917039125963949, −5.511697059692515, −4.717406191870146, −4.182946410794027, −3.311399742594830, −2.685907003957964, −2.055752046892085, −0.9525648281292886, 0, 0.9525648281292886, 2.055752046892085, 2.685907003957964, 3.311399742594830, 4.182946410794027, 4.717406191870146, 5.511697059692515, 5.917039125963949, 6.880464833966881, 7.056496054815601, 7.950512348243569, 8.419007952844940, 9.198843671572595, 9.378800090854644, 10.37693478545647, 10.74191564527821, 11.18340640183898, 11.77244404368700, 12.61195127029303, 13.02234060774735, 13.43843896836280, 13.76149942117291, 14.73811365764453, 15.27000594480363, 15.45459671355163

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