Properties

Degree $4$
Conductor $142884$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 6·5-s − 7-s + 16-s + 6·20-s + 17·25-s − 28-s − 6·35-s + 4·37-s − 12·41-s − 20·43-s + 12·47-s − 6·49-s + 24·59-s + 64-s + 28·67-s + 16·79-s + 6·80-s − 6·83-s − 36·89-s + 17·100-s − 6·101-s + 4·109-s − 112-s − 13·121-s + 18·125-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s + 2.68·5-s − 0.377·7-s + 1/4·16-s + 1.34·20-s + 17/5·25-s − 0.188·28-s − 1.01·35-s + 0.657·37-s − 1.87·41-s − 3.04·43-s + 1.75·47-s − 6/7·49-s + 3.12·59-s + 1/8·64-s + 3.42·67-s + 1.80·79-s + 0.670·80-s − 0.658·83-s − 3.81·89-s + 1.69·100-s − 0.597·101-s + 0.383·109-s − 0.0944·112-s − 1.18·121-s + 1.60·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(142884\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{142884} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 142884,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.900027461\)
\(L(\frac12)\) \(\approx\) \(2.900027461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.507801646196580209679416383095, −9.022301181159976542679631550139, −8.268048909391662018895990050852, −8.153752740732516759811873175243, −6.90390849159694968636551446179, −6.73860900674537002760989556093, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.37363015537505503550836847886, −5.14293667469015758909535382312, −4.01126047180402972127592124525, −3.26671376711039705407751493003, −2.44850978333263328730546301919, −2.07276435338156744028852711093, −1.33346620954041525748246641205, 1.33346620954041525748246641205, 2.07276435338156744028852711093, 2.44850978333263328730546301919, 3.26671376711039705407751493003, 4.01126047180402972127592124525, 5.14293667469015758909535382312, 5.37363015537505503550836847886, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.73860900674537002760989556093, 6.90390849159694968636551446179, 8.153752740732516759811873175243, 8.268048909391662018895990050852, 9.022301181159976542679631550139, 9.507801646196580209679416383095

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