Properties

Degree $2$
Conductor $235200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·11-s + 2·13-s + 4·17-s − 8·23-s − 27-s − 2·31-s + 2·33-s + 8·37-s − 2·39-s − 2·41-s − 2·43-s − 10·47-s − 4·51-s − 2·53-s − 4·59-s + 10·61-s + 2·67-s + 8·69-s − 12·71-s − 10·73-s + 16·79-s + 81-s + 16·83-s + 14·89-s + 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.66·23-s − 0.192·27-s − 0.359·31-s + 0.348·33-s + 1.31·37-s − 0.320·39-s − 0.312·41-s − 0.304·43-s − 1.45·47-s − 0.560·51-s − 0.274·53-s − 0.520·59-s + 1.28·61-s + 0.244·67-s + 0.963·69-s − 1.42·71-s − 1.17·73-s + 1.80·79-s + 1/9·81-s + 1.75·83-s + 1.48·89-s + 0.207·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613436832\)
\(L(\frac12)\) \(\approx\) \(1.613436832\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.83837059518807, −12.47583625898524, −11.89417666464088, −11.52932036848549, −11.18813249392469, −10.44011019746679, −10.22443563503375, −9.796241018072523, −9.261307453299981, −8.619650577992081, −8.120501159959119, −7.663074594868605, −7.404824562574366, −6.486885051923199, −6.194238127401514, −5.809601596023573, −5.182586197742359, −4.754788828065466, −4.177258877745486, −3.474807742399971, −3.222112079453033, −2.226844325810907, −1.856650997964227, −1.034073764739358, −0.4082005529925504, 0.4082005529925504, 1.034073764739358, 1.856650997964227, 2.226844325810907, 3.222112079453033, 3.474807742399971, 4.177258877745486, 4.754788828065466, 5.182586197742359, 5.809601596023573, 6.194238127401514, 6.486885051923199, 7.404824562574366, 7.663074594868605, 8.120501159959119, 8.619650577992081, 9.261307453299981, 9.796241018072523, 10.22443563503375, 10.44011019746679, 11.18813249392469, 11.52932036848549, 11.89417666464088, 12.47583625898524, 12.83837059518807

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