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Properties

Label 4-20e2-1.1-c27e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $8532.42$
Root an. cond. $9.61098$
Motivic weight $27$
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  + 1.63e4·2-s + 1.34e8·4-s − 5.45e9·5-s − 8.93e13·10-s + 2.51e15·13-s − 1.80e16·16-s + 1.03e17·17-s − 7.31e17·20-s + 2.22e19·25-s + 4.12e19·26-s − 2.95e20·32-s + 1.70e21·34-s − 2.12e21·37-s − 1.39e22·41-s + 3.65e23·50-s + 3.38e23·52-s − 2.31e23·53-s + 5.03e24·61-s − 2.41e24·64-s − 1.37e25·65-s + 1.39e25·68-s + 2.01e25·73-s − 3.48e25·74-s + 9.82e25·80-s − 5.81e25·81-s − 2.29e26·82-s − 5.65e26·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.99·5-s − 2.82·10-s + 2.30·13-s − 16-s + 2.54·17-s − 1.99·20-s + 2.99·25-s + 3.26·26-s − 1.41·32-s + 3.59·34-s − 1.43·37-s − 2.36·41-s + 4.22·50-s + 2.30·52-s − 1.21·53-s + 3.98·61-s − 64-s − 4.60·65-s + 2.54·68-s + 1.41·73-s − 2.02·74-s + 1.99·80-s − 81-s − 3.33·82-s − 5.07·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(8532.42\)
Root analytic conductor: \(9.61098\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(5.222953770\)
\(L(\frac12)\) \(\approx\) \(5.222953770\)
\(L(\frac{29}{2})\) 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$C_2$ \( 1 - p^{14} T + p^{27} T^{2} \)
5$C_2$ \( 1 + 5452892644 T + p^{27} T^{2} \)
good3$C_2^2$ \( 1 + p^{54} T^{4} \)
7$C_2^2$ \( 1 + p^{54} T^{4} \)
11$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2153275275509298 T + p^{27} T^{2} )( 1 - 365429552623108 T + p^{27} T^{2} ) \)
17$C_2$ \( ( 1 - 77235397780956584 T + p^{27} T^{2} )( 1 - 26561976927408094 T + p^{27} T^{2} ) \)
19$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{54} T^{4} \)
29$C_2$ \( ( 1 - 73026732471353912030 T + p^{27} T^{2} )( 1 + 73026732471353912030 T + p^{27} T^{2} ) \)
31$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
37$C_2$ \( ( 1 - \)\(74\!\cdots\!84\)\( T + p^{27} T^{2} )( 1 + \)\(28\!\cdots\!26\)\( T + p^{27} T^{2} ) \)
41$C_2$ \( ( 1 + \)\(69\!\cdots\!68\)\( T + p^{27} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{54} T^{4} \)
47$C_2^2$ \( 1 + p^{54} T^{4} \)
53$C_2$ \( ( 1 - \)\(12\!\cdots\!98\)\( T + p^{27} T^{2} )( 1 + \)\(35\!\cdots\!12\)\( T + p^{27} T^{2} ) \)
59$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(25\!\cdots\!72\)\( T + p^{27} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{54} T^{4} \)
71$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(27\!\cdots\!08\)\( T + p^{27} T^{2} )( 1 + \)\(74\!\cdots\!82\)\( T + p^{27} T^{2} ) \)
79$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{54} T^{4} \)
89$C_2$ \( ( 1 - \)\(24\!\cdots\!30\)\( T + p^{27} T^{2} )( 1 + \)\(24\!\cdots\!30\)\( T + p^{27} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(12\!\cdots\!04\)\( T + p^{27} T^{2} )( 1 - \)\(38\!\cdots\!94\)\( T + p^{27} T^{2} ) \)
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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

−12.84465279120821072312807284172, −12.13867502736498024237363184222, −11.85983054578740177818927973064, −11.29814978529990639700874350579, −10.73825098576379340921712821599, −9.922261966669807362463755118754, −8.707079735893600507653966121988, −8.361669541934798047656475808975, −7.82157332379915752069497675527, −6.89852775105504487302880500051, −6.52178652608826365133560245929, −5.38308433839240378402683795559, −5.30097554868097029286340128314, −4.22189007769823738233253473680, −3.72866275487944365147421372911, −3.33523803403668149364138408136, −3.15695392730594207340677174139, −1.74595450503092812979284697822, −0.996628189867761726358163342778, −0.48105483028397220826710428687, 0.48105483028397220826710428687, 0.996628189867761726358163342778, 1.74595450503092812979284697822, 3.15695392730594207340677174139, 3.33523803403668149364138408136, 3.72866275487944365147421372911, 4.22189007769823738233253473680, 5.30097554868097029286340128314, 5.38308433839240378402683795559, 6.52178652608826365133560245929, 6.89852775105504487302880500051, 7.82157332379915752069497675527, 8.361669541934798047656475808975, 8.707079735893600507653966121988, 9.922261966669807362463755118754, 10.73825098576379340921712821599, 11.29814978529990639700874350579, 11.85983054578740177818927973064, 12.13867502736498024237363184222, 12.84465279120821072312807284172

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