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 − 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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
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

−13.23281575835588, −12.67167052453016, −12.24073842323712, −11.90097566051323, −11.13977175658089, −10.89210684549128, −10.26752439809428, −9.793182020667895, −9.522896245332100, −8.880843870261184, −8.425010262395249, −7.942680027283553, −7.601467023896930, −7.045212948111337, −6.511041642011971, −5.936399035964928, −5.527529459202557, −4.788706126407822, −4.253658488905882, −4.046774907853584, −3.144557581982282, −2.660092466678644, −2.203098851582523, −1.672506268933128, −0.7321271636569396, 0, 0.7321271636569396, 1.672506268933128, 2.203098851582523, 2.660092466678644, 3.144557581982282, 4.046774907853584, 4.253658488905882, 4.788706126407822, 5.527529459202557, 5.936399035964928, 6.511041642011971, 7.045212948111337, 7.601467023896930, 7.942680027283553, 8.425010262395249, 8.880843870261184, 9.522896245332100, 9.793182020667895, 10.26752439809428, 10.89210684549128, 11.13977175658089, 11.90097566051323, 12.24073842323712, 12.67167052453016, 13.23281575835588

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