Properties

Label 4-20e4-1.1-c5e2-0-6
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  + 8·3-s − 8·7-s − 197·9-s + 200·11-s + 592·13-s − 278·17-s + 840·19-s − 64·21-s − 1.95e3·23-s − 1.72e3·27-s − 4.68e3·29-s + 5.00e3·31-s + 1.60e3·33-s − 1.25e4·37-s + 4.73e3·39-s − 5.33e3·41-s + 224·43-s + 2.60e4·47-s − 3.26e4·49-s − 2.22e3·51-s + 4.68e4·53-s + 6.72e3·57-s + 8.17e4·59-s − 4.69e4·61-s + 1.57e3·63-s + 6.88e4·67-s − 1.56e4·69-s + ⋯
L(s)  = 1  + 0.513·3-s − 0.0617·7-s − 0.810·9-s + 0.498·11-s + 0.971·13-s − 0.233·17-s + 0.533·19-s − 0.0316·21-s − 0.769·23-s − 0.454·27-s − 1.03·29-s + 0.935·31-s + 0.255·33-s − 1.50·37-s + 0.498·39-s − 0.495·41-s + 0.0184·43-s + 1.72·47-s − 1.93·49-s − 0.119·51-s + 2.28·53-s + 0.273·57-s + 3.05·59-s − 1.61·61-s + 0.0500·63-s + 1.87·67-s − 0.394·69-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: induced by $\chi_{400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.609513206\)
\(L(\frac12)\) \(\approx\) \(3.609513206\)
\(L(\frac{7}{2})\) 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$D_{4}$ \( 1 - 8 T + 29 p^{2} T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 8 T + 32666 T^{2} + 8 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 200 T + 225821 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 592 T + 178538 T^{2} - 592 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 278 T + 2797339 T^{2} + 278 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 840 T + 4289677 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 1952 T + 11510698 T^{2} + 1952 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 4680 T + 1421346 p T^{2} + 4680 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 5008 T + 33327162 T^{2} - 5008 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 12500 T + 121294718 T^{2} + 12500 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 5334 T + 27686155 T^{2} + 5334 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 224 T + 15005414 T^{2} - 224 p^{5} T^{3} + p^{10} T^{4} \)
47$C_2$ \( ( 1 - 13036 T + p^{5} T^{2} )^{2} \)
53$D_{4}$ \( 1 - 46812 T + 1352998222 T^{2} - 46812 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 81776 T + 3059970646 T^{2} - 81776 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 46932 T + 2239379182 T^{2} + 46932 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 68808 T + 3858742141 T^{2} - 68808 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 7448 T + 3593807902 T^{2} + 7448 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 108822 T + 7105450763 T^{2} - 108822 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 108104 T + 6816153098 T^{2} - 108104 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 328 p T + 4507827109 T^{2} - 328 p^{6} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 70990 T + 6345035107 T^{2} - 70990 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 96852 T + 16039717990 T^{2} - 96852 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.56756038960455007365553590237, −10.31428617868854528135934790250, −9.674978240046098165215528507630, −9.277151883823512591451170063467, −8.665656914040454513598236539186, −8.610956995624248792019258833458, −7.975404600083867171843554572149, −7.58912611084731348858165391768, −6.70196640617827131516589353408, −6.65945158157963821284041533125, −5.86727311680305987173083706130, −5.45226657850375047959547540479, −4.95556458544384575079629690862, −4.08242977034811974254808353409, −3.52189963238889718297822664588, −3.40816096822972329998766620377, −2.25612514199185473571426732836, −2.11659886807368425994540263481, −1.03532475423679087487587356143, −0.51508641560616226091748210157, 0.51508641560616226091748210157, 1.03532475423679087487587356143, 2.11659886807368425994540263481, 2.25612514199185473571426732836, 3.40816096822972329998766620377, 3.52189963238889718297822664588, 4.08242977034811974254808353409, 4.95556458544384575079629690862, 5.45226657850375047959547540479, 5.86727311680305987173083706130, 6.65945158157963821284041533125, 6.70196640617827131516589353408, 7.58912611084731348858165391768, 7.975404600083867171843554572149, 8.610956995624248792019258833458, 8.665656914040454513598236539186, 9.277151883823512591451170063467, 9.674978240046098165215528507630, 10.31428617868854528135934790250, 10.56756038960455007365553590237

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