Properties

Label 4-20e4-1.1-c5e2-0-10
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $4115.67$
Root an. cond. $8.00958$
Motivic weight $5$
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  − 950·13-s + 3.05e3·17-s + 950·37-s − 9.90e3·41-s + 3.29e4·53-s + 1.09e5·61-s + 1.08e5·73-s − 5.90e4·81-s + 2.52e5·97-s + 1.96e5·101-s + 3.62e5·113-s + 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.51e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.55·13-s + 2.55·17-s + 0.114·37-s − 0.920·41-s + 1.61·53-s + 3.78·61-s + 2.39·73-s − 81-s + 2.72·97-s + 1.91·101-s + 2.67·113-s + 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.21·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4115.67\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.697526294\)
\(L(\frac12)\) \(\approx\) \(3.697526294\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{10} T^{4} \)
7$C_2^2$ \( 1 + p^{10} T^{4} \)
11$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 244 T + p^{5} T^{2} )( 1 + 1194 T + p^{5} T^{2} ) \)
17$C_2$ \( ( 1 - 2242 T + p^{5} T^{2} )( 1 - 808 T + p^{5} T^{2} ) \)
19$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 2950 T + p^{5} T^{2} )( 1 + 2950 T + p^{5} T^{2} ) \)
31$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12242 T + p^{5} T^{2} )( 1 + 11292 T + p^{5} T^{2} ) \)
41$C_2$ \( ( 1 + 4952 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{10} T^{4} \)
47$C_2^2$ \( 1 + p^{10} T^{4} \)
53$C_2$ \( ( 1 - 40244 T + p^{5} T^{2} )( 1 + 7294 T + p^{5} T^{2} ) \)
59$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 54948 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{10} T^{4} \)
71$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 88806 T + p^{5} T^{2} )( 1 - 20144 T + p^{5} T^{2} ) \)
79$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 51050 T + p^{5} T^{2} )( 1 + 51050 T + p^{5} T^{2} ) \)
97$C_2$ \( ( 1 - 160808 T + p^{5} T^{2} )( 1 - 92142 T + p^{5} T^{2} ) \)
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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

−10.42202382405868765536864122734, −10.14352961324209924308368355977, −9.810105413294988594384792175882, −9.626437047779028774656895765353, −8.639243976711324484461782157148, −8.549996342411907806984742529465, −7.66823918790390842642178553563, −7.60133216384385828277995254982, −7.02072544485713334750403221353, −6.53380084246607686439971068467, −5.70834866217376372879384064520, −5.42423730894994288232472039899, −4.99108206832676203563115094615, −4.35518388093080463352917965646, −3.42160442214411884183110652868, −3.39055212162405866195969382203, −2.35210573542416868815274846457, −1.99480937220906664094910365709, −0.860250705980732214930134167915, −0.63185188417247540858696777493, 0.63185188417247540858696777493, 0.860250705980732214930134167915, 1.99480937220906664094910365709, 2.35210573542416868815274846457, 3.39055212162405866195969382203, 3.42160442214411884183110652868, 4.35518388093080463352917965646, 4.99108206832676203563115094615, 5.42423730894994288232472039899, 5.70834866217376372879384064520, 6.53380084246607686439971068467, 7.02072544485713334750403221353, 7.60133216384385828277995254982, 7.66823918790390842642178553563, 8.549996342411907806984742529465, 8.639243976711324484461782157148, 9.626437047779028774656895765353, 9.810105413294988594384792175882, 10.14352961324209924308368355977, 10.42202382405868765536864122734

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