Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·11-s + 3·13-s + 4·17-s + 5·19-s − 4·23-s − 27-s + 4·29-s + 7·31-s + 6·33-s − 9·37-s − 3·39-s + 2·41-s + 43-s − 2·47-s − 4·51-s + 8·53-s − 5·57-s + 10·61-s + 15·67-s + 4·69-s + 6·71-s − 11·73-s − 79-s + 81-s + 6·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.832·13-s + 0.970·17-s + 1.14·19-s − 0.834·23-s − 0.192·27-s + 0.742·29-s + 1.25·31-s + 1.04·33-s − 1.47·37-s − 0.480·39-s + 0.312·41-s + 0.152·43-s − 0.291·47-s − 0.560·51-s + 1.09·53-s − 0.662·57-s + 1.28·61-s + 1.83·67-s + 0.481·69-s + 0.712·71-s − 1.28·73-s − 0.112·79-s + 1/9·81-s + 0.658·83-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: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−13.08594123103851, −12.65701636175832, −12.17381581453363, −11.76426390537285, −11.35090136598941, −10.73125549171060, −10.35737783556253, −9.941893910204120, −9.735738085443537, −8.766989935629196, −8.398316838821156, −7.925108833090195, −7.543149019412546, −6.971395300128299, −6.431958500258137, −5.794464664548280, −5.493013832653120, −5.031594232148521, −4.588627194132721, −3.661164720916289, −3.455142205071367, −2.619184163292064, −2.236694853013275, −1.247688267532819, −0.8298708823778343, 0, 0.8298708823778343, 1.247688267532819, 2.236694853013275, 2.619184163292064, 3.455142205071367, 3.661164720916289, 4.588627194132721, 5.031594232148521, 5.493013832653120, 5.794464664548280, 6.431958500258137, 6.971395300128299, 7.543149019412546, 7.925108833090195, 8.398316838821156, 8.766989935629196, 9.735738085443537, 9.941893910204120, 10.35737783556253, 10.73125549171060, 11.35090136598941, 11.76426390537285, 12.17381581453363, 12.65701636175832, 13.08594123103851

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