Properties

Label 4-228e2-1.1-c7e2-0-2
Degree $4$
Conductor $51984$
Sign $1$
Analytic cond. $5072.82$
Root an. cond. $8.43941$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s + 2.51e3·7-s + 4.37e3·9-s + 1.66e4·13-s + 5.74e4·19-s + 2.03e5·21-s − 7.81e4·25-s + 1.77e5·27-s + 1.34e6·39-s + 6.25e5·43-s + 3.07e6·49-s + 4.65e6·57-s + 1.53e6·61-s + 1.09e7·63-s + 3.67e6·67-s − 5.03e6·73-s − 6.32e6·75-s − 1.32e7·79-s + 4.78e6·81-s + 4.17e7·91-s + 2.27e7·97-s + 3.66e7·109-s + 7.27e7·117-s + 3.89e7·121-s + 127-s + 5.06e7·129-s + 131-s + ⋯
L(s)  = 1  + 1.73·3-s + 2.76·7-s + 2·9-s + 2.09·13-s + 1.92·19-s + 4.79·21-s − 25-s + 1.73·27-s + 3.63·39-s + 1.20·43-s + 3.73·49-s + 3.32·57-s + 0.867·61-s + 5.53·63-s + 1.49·67-s − 1.51·73-s − 1.73·75-s − 3.03·79-s + 81-s + 5.80·91-s + 2.53·97-s + 2.71·109-s + 4.19·117-s + 2·121-s + 2.07·129-s + 5.31·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51984\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5072.82\)
Root analytic conductor: \(8.43941\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 51984,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(16.37705136\)
\(L(\frac12)\) \(\approx\) \(16.37705136\)
\(L(\frac{9}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p^{4} T + p^{7} T^{2} \)
19$C_2$ \( 1 - 57448 T + p^{7} T^{2} \)
good5$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
7$C_2$ \( ( 1 - 1255 T + p^{7} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 14614 T + p^{7} T^{2} )( 1 - 2009 T + p^{7} T^{2} ) \)
17$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
23$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
29$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
31$C_2$ \( ( 1 - 331387 T + p^{7} T^{2} )( 1 + 331387 T + p^{7} T^{2} ) \)
37$C_2$ \( ( 1 - 615373 T + p^{7} T^{2} )( 1 + 615373 T + p^{7} T^{2} ) \)
41$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
43$C_2$ \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 409495 T + p^{7} T^{2} ) \)
47$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
53$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
59$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
61$C_2$ \( ( 1 - 3535546 T + p^{7} T^{2} )( 1 + 1998347 T + p^{7} T^{2} ) \)
67$C_2$ \( ( 1 - 4058455 T + p^{7} T^{2} )( 1 + 385072 T + p^{7} T^{2} ) \)
71$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
73$C_2$ \( ( 1 - 1236809 T + p^{7} T^{2} )( 1 + 6274810 T + p^{7} T^{2} ) \)
79$C_2$ \( ( 1 + 4517617 T + p^{7} T^{2} )( 1 + 8763044 T + p^{7} T^{2} ) \)
83$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
97$C_2$ \( ( 1 - 17521555 T + p^{7} T^{2} )( 1 - 5276357 T + p^{7} T^{2} ) \)
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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

−11.12670129602436502736539843231, −10.92006394749158283474931522813, −10.07814082815299494919386739317, −9.725284354648494192104921963516, −8.872310784136948847829934545954, −8.750921197861364784095573851122, −8.194690946695603968525882553913, −7.992765912240270090207913839765, −7.37598874187988287821913269904, −7.15707666437944582989410699415, −5.83514891995121668254735819137, −5.67478273476357126248123643159, −4.63708706667155495977292944274, −4.46923315721935421739553241972, −3.58793338379008295952641497385, −3.33623631310703067231664803399, −2.28297273523005233342013991231, −1.87383393585098918442768792571, −1.13937905260409362533743617725, −1.08798973396424355868139893663, 1.08798973396424355868139893663, 1.13937905260409362533743617725, 1.87383393585098918442768792571, 2.28297273523005233342013991231, 3.33623631310703067231664803399, 3.58793338379008295952641497385, 4.46923315721935421739553241972, 4.63708706667155495977292944274, 5.67478273476357126248123643159, 5.83514891995121668254735819137, 7.15707666437944582989410699415, 7.37598874187988287821913269904, 7.992765912240270090207913839765, 8.194690946695603968525882553913, 8.750921197861364784095573851122, 8.872310784136948847829934545954, 9.725284354648494192104921963516, 10.07814082815299494919386739317, 10.92006394749158283474931522813, 11.12670129602436502736539843231

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