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Properties

Label 4-20e2-1.1-c26e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $7337.39$
Root an. cond. $9.25519$
Motivic weight $26$
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  − 6.71e7·4-s + 2.12e9·5-s − 5.08e12·9-s + 4.50e15·16-s − 1.42e17·20-s + 3.04e18·25-s − 3.66e19·29-s + 3.41e20·36-s − 9.67e20·41-s − 1.08e22·45-s − 1.87e22·49-s + 4.57e23·61-s − 3.02e23·64-s + 9.58e24·80-s + 1.93e25·81-s + 3.31e25·89-s − 2.04e26·100-s + 3.88e26·101-s + 3.34e26·109-s + 2.45e27·116-s + 2.38e27·121-s + 3.30e27·125-s + 127-s + 131-s + 137-s + 139-s − 2.28e28·144-s + ⋯
L(s)  = 1  − 4-s + 1.74·5-s − 2·9-s + 16-s − 1.74·20-s + 2.04·25-s − 3.56·29-s + 2·36-s − 1.04·41-s − 3.48·45-s − 2·49-s + 2.82·61-s − 64-s + 1.74·80-s + 3·81-s + 1.50·89-s − 2.04·100-s + 3.40·101-s + 1.09·109-s + 3.56·116-s + 2·121-s + 1.81·125-s − 2·144-s − 6.22·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+13)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(7337.39\)
Root analytic conductor: \(9.25519\)
Motivic weight: \(26\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :13, 13),\ 1)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.6888037010\)
\(L(\frac12)\) \(\approx\) \(0.6888037010\)
\(L(14)\) 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$C_2$ \( 1 + p^{26} T^{2} \)
5$C_2$ \( 1 - 2128894566 T + p^{26} T^{2} \)
good3$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
7$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
13$C_2$ \( ( 1 - 553129610136470 T + p^{26} T^{2} )( 1 + 553129610136470 T + p^{26} T^{2} ) \)
17$C_2$ \( ( 1 - 19735504003013790 T + p^{26} T^{2} )( 1 + 19735504003013790 T + p^{26} T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
23$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 18311061619791551478 T + p^{26} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
37$C_2$ \( ( 1 - \)\(19\!\cdots\!90\)\( T + p^{26} T^{2} )( 1 + \)\(19\!\cdots\!90\)\( T + p^{26} T^{2} ) \)
41$C_2$ \( ( 1 + \)\(48\!\cdots\!58\)\( T + p^{26} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
53$C_2$ \( ( 1 - \)\(30\!\cdots\!70\)\( T + p^{26} T^{2} )( 1 + \)\(30\!\cdots\!70\)\( T + p^{26} T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
61$C_2$ \( ( 1 - \)\(22\!\cdots\!62\)\( T + p^{26} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
73$C_2$ \( ( 1 - \)\(33\!\cdots\!70\)\( T + p^{26} T^{2} )( 1 + \)\(33\!\cdots\!70\)\( T + p^{26} T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
83$C_2$ \( ( 1 + p^{26} T^{2} )^{2} \)
89$C_2$ \( ( 1 - \)\(16\!\cdots\!62\)\( T + p^{26} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(16\!\cdots\!10\)\( T + p^{26} T^{2} )( 1 + \)\(16\!\cdots\!10\)\( T + p^{26} T^{2} ) \)
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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

−13.02681638419168374027606462858, −12.79190557203442658327542233141, −11.52241884560838747513133036260, −11.24843007091099260597502962696, −10.31652730409471798569760681960, −9.709277617800466560561363605990, −9.217258545971607771544671870814, −8.765179080480085066047857025001, −8.172611198252988747211970354491, −7.26515952793395573147742611805, −6.14361340364384018364435209640, −5.90959472764858833121941873152, −5.22314683953893893526229555019, −4.97004335898282435074000182172, −3.56704996180535892234494097545, −3.33853466617513055652880127556, −2.21539896068817772311030610611, −1.99375059352598424132046388259, −1.02602712166264481036208016643, −0.20547198371155278547895858391, 0.20547198371155278547895858391, 1.02602712166264481036208016643, 1.99375059352598424132046388259, 2.21539896068817772311030610611, 3.33853466617513055652880127556, 3.56704996180535892234494097545, 4.97004335898282435074000182172, 5.22314683953893893526229555019, 5.90959472764858833121941873152, 6.14361340364384018364435209640, 7.26515952793395573147742611805, 8.172611198252988747211970354491, 8.765179080480085066047857025001, 9.217258545971607771544671870814, 9.709277617800466560561363605990, 10.31652730409471798569760681960, 11.24843007091099260597502962696, 11.52241884560838747513133036260, 12.79190557203442658327542233141, 13.02681638419168374027606462858

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