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Properties

Label 4-20e2-1.1-c43e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $54859.3$
Root an. cond. $15.3042$
Motivic weight $43$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.19e6·2-s + 8.79e12·4-s − 9.91e14·5-s − 4.15e21·10-s − 2.02e24·13-s − 7.73e25·16-s − 7.63e26·17-s − 8.72e27·20-s − 1.54e29·25-s − 8.51e30·26-s − 3.24e32·32-s − 3.20e33·34-s + 4.82e32·37-s − 1.27e35·41-s − 6.46e35·50-s − 1.78e37·52-s + 3.28e37·53-s + 2.77e37·61-s − 6.80e38·64-s + 2.01e39·65-s − 6.71e39·68-s + 2.85e40·73-s + 2.02e39·74-s + 7.67e40·80-s − 1.07e41·81-s − 5.34e41·82-s + 7.57e41·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.929·5-s − 1.31·10-s − 2.27·13-s − 16-s − 2.68·17-s − 0.929·20-s − 0.135·25-s − 3.22·26-s − 1.41·32-s − 3.79·34-s + 0.0927·37-s − 2.69·41-s − 0.191·50-s − 2.27·52-s + 2.77·53-s + 0.114·61-s − 64-s + 2.11·65-s − 2.68·68-s + 2.47·73-s + 0.131·74-s + 0.929·80-s − 81-s − 3.81·82-s + 2.49·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(54859.3\)
Root analytic conductor: \(15.3042\)
Motivic weight: \(43\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :43/2, 43/2),\ 1)\)

Particular Values

\(L(22)\) \(\approx\) \(2.535434148\)
\(L(\frac12)\) \(\approx\) \(2.535434148\)
\(L(\frac{45}{2})\) 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$C_2$ \( 1 - p^{22} T + p^{43} T^{2} \)
5$C_2$ \( 1 + 991358892314404 T + p^{43} T^{2} \)
good3$C_2^2$ \( 1 + p^{86} T^{4} \)
7$C_2^2$ \( 1 + p^{86} T^{4} \)
11$C_2$ \( ( 1 - p^{43} T^{2} )^{2} \)
13$C_2$ \( ( 1 + \)\(26\!\cdots\!92\)\( T + p^{43} T^{2} )( 1 + \)\(17\!\cdots\!82\)\( T + p^{43} T^{2} ) \)
17$C_2$ \( ( 1 + \)\(25\!\cdots\!96\)\( T + p^{43} T^{2} )( 1 + \)\(51\!\cdots\!06\)\( T + p^{43} T^{2} ) \)
19$C_2$ \( ( 1 + p^{43} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{86} T^{4} \)
29$C_2$ \( ( 1 - \)\(43\!\cdots\!70\)\( T + p^{43} T^{2} )( 1 + \)\(43\!\cdots\!70\)\( T + p^{43} T^{2} ) \)
31$C_2$ \( ( 1 - p^{43} T^{2} )^{2} \)
37$C_2$ \( ( 1 - \)\(75\!\cdots\!14\)\( T + p^{43} T^{2} )( 1 + \)\(71\!\cdots\!96\)\( T + p^{43} T^{2} ) \)
41$C_2$ \( ( 1 + \)\(63\!\cdots\!28\)\( T + p^{43} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{86} T^{4} \)
47$C_2^2$ \( 1 + p^{86} T^{4} \)
53$C_2$ \( ( 1 - \)\(19\!\cdots\!18\)\( T + p^{43} T^{2} )( 1 - \)\(13\!\cdots\!28\)\( T + p^{43} T^{2} ) \)
59$C_2$ \( ( 1 + p^{43} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(13\!\cdots\!32\)\( T + p^{43} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{86} T^{4} \)
71$C_2$ \( ( 1 - p^{43} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(22\!\cdots\!08\)\( T + p^{43} T^{2} )( 1 - \)\(63\!\cdots\!98\)\( T + p^{43} T^{2} ) \)
79$C_2$ \( ( 1 + p^{43} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{86} T^{4} \)
89$C_2$ \( ( 1 - \)\(14\!\cdots\!30\)\( T + p^{43} T^{2} )( 1 + \)\(14\!\cdots\!30\)\( T + p^{43} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(79\!\cdots\!84\)\( T + p^{43} T^{2} )( 1 - \)\(67\!\cdots\!94\)\( T + p^{43} T^{2} ) \)
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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

−11.56980823243661541391268265906, −10.65560989004007682582355598429, −10.08346049600369159750197283871, −9.361902544753439794031811890485, −8.743381905995982889596336638706, −8.296700214938050492610279626176, −7.29791699278291879616058175381, −7.10421005340101664624277607436, −6.60980682472538822269085742777, −5.91577534673407228269679103890, −5.09265446760925633863187929959, −4.76983468250537341033206676164, −4.48940300706401528878809110847, −3.78500785659529934415400781026, −3.41099815533036412692348384300, −2.61171055932800958595899114543, −2.09960995184503628215872471556, −2.02560629463677750007012580316, −0.49335576581568862514339288164, −0.38193756782734734491639803738, 0.38193756782734734491639803738, 0.49335576581568862514339288164, 2.02560629463677750007012580316, 2.09960995184503628215872471556, 2.61171055932800958595899114543, 3.41099815533036412692348384300, 3.78500785659529934415400781026, 4.48940300706401528878809110847, 4.76983468250537341033206676164, 5.09265446760925633863187929959, 5.91577534673407228269679103890, 6.60980682472538822269085742777, 7.10421005340101664624277607436, 7.29791699278291879616058175381, 8.296700214938050492610279626176, 8.743381905995982889596336638706, 9.361902544753439794031811890485, 10.08346049600369159750197283871, 10.65560989004007682582355598429, 11.56980823243661541391268265906

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