Properties

Label 4-200e2-1.1-c5e2-0-6
Degree $4$
Conductor $40000$
Sign $1$
Analytic cond. $1028.91$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40000\)    =    \(2^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1028.91\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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

−11.21273078240248233241254584625, −10.96227984832216863275497415161, −10.32597659820493160490778207021, −9.776283672926727092686449226281, −9.234280630610671286140231521472, −9.030370538735780683620292940981, −8.136392878723981349970843941112, −7.84900020054086527388735682328, −7.47807726339863229060292843362, −6.69145796210014827263024826897, −5.88859056451622014673679419598, −5.75438696283865101615989628798, −4.73235305893953537125787853473, −4.41390048360393067481001438311, −3.34106044768812835192819469995, −2.97089029988153106957745387557, −2.20369745859482580075446935401, −1.53111696703952827659537950607, 0, 0, 1.53111696703952827659537950607, 2.20369745859482580075446935401, 2.97089029988153106957745387557, 3.34106044768812835192819469995, 4.41390048360393067481001438311, 4.73235305893953537125787853473, 5.75438696283865101615989628798, 5.88859056451622014673679419598, 6.69145796210014827263024826897, 7.47807726339863229060292843362, 7.84900020054086527388735682328, 8.136392878723981349970843941112, 9.030370538735780683620292940981, 9.234280630610671286140231521472, 9.776283672926727092686449226281, 10.32597659820493160490778207021, 10.96227984832216863275497415161, 11.21273078240248233241254584625

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