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Properties

Label 4-20e2-1.1-c1e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $0.0255043$
Root an. cond. $0.399625$
Motivic weight $1$
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  − 2·2-s + 2·4-s − 4·5-s + 8·10-s − 2·13-s − 4·16-s + 6·17-s − 8·20-s + 11·25-s + 4·26-s + 8·32-s − 12·34-s − 14·37-s − 16·41-s − 22·50-s − 4·52-s + 18·53-s + 24·61-s − 8·64-s + 8·65-s + 12·68-s − 22·73-s + 28·74-s + 16·80-s − 9·81-s + 32·82-s − 24·85-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s + 2.52·10-s − 0.554·13-s − 16-s + 1.45·17-s − 1.78·20-s + 11/5·25-s + 0.784·26-s + 1.41·32-s − 2.05·34-s − 2.30·37-s − 2.49·41-s − 3.11·50-s − 0.554·52-s + 2.47·53-s + 3.07·61-s − 64-s + 0.992·65-s + 1.45·68-s − 2.57·73-s + 3.25·74-s + 1.78·80-s − 81-s + 3.53·82-s − 2.60·85-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.0255043\)
Root analytic conductor: \(0.399625\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1839085544\)
\(L(\frac12)\) \(\approx\) \(0.1839085544\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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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

−18.63950642032327279407698589487, −18.54530994742234557988936814298, −17.36278766896153774645413833010, −17.05197663834121013206113169629, −16.18889154393944071424592053479, −15.98851315466574025753781465568, −15.12087754482813165213151877280, −14.60863390629778476709057892335, −13.61776821454158890200258592321, −12.57162116369725768064984496401, −11.72375060446644028152469300011, −11.62341391771111472288252712239, −10.22889472215849124414124733521, −10.19606565450750423970831368888, −8.640379848934142593144264292186, −8.476557285997462789360043827500, −7.36427401884056470781626500805, −7.08937157793967758816517109516, −5.09388682361178306352765485359, −3.63351761276323569547339575571, 3.63351761276323569547339575571, 5.09388682361178306352765485359, 7.08937157793967758816517109516, 7.36427401884056470781626500805, 8.476557285997462789360043827500, 8.640379848934142593144264292186, 10.19606565450750423970831368888, 10.22889472215849124414124733521, 11.62341391771111472288252712239, 11.72375060446644028152469300011, 12.57162116369725768064984496401, 13.61776821454158890200258592321, 14.60863390629778476709057892335, 15.12087754482813165213151877280, 15.98851315466574025753781465568, 16.18889154393944071424592053479, 17.05197663834121013206113169629, 17.36278766896153774645413833010, 18.54530994742234557988936814298, 18.63950642032327279407698589487

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