Properties

Label 4-80e2-1.1-c9e2-0-2
Degree $4$
Conductor $6400$
Sign $1$
Analytic cond. $1697.67$
Root an. cond. $6.41894$
Motivic weight $9$
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  + 92·3-s + 1.25e3·5-s + 6.90e3·7-s − 8.82e3·9-s + 8.08e3·11-s + 1.11e5·13-s + 1.15e5·15-s − 3.27e5·17-s + 1.15e6·19-s + 6.35e5·21-s + 1.05e6·23-s + 1.17e6·25-s − 5.91e5·27-s − 4.21e6·29-s + 1.13e7·31-s + 7.43e5·33-s + 8.63e6·35-s − 7.25e6·37-s + 1.02e7·39-s + 1.30e7·41-s + 4.79e7·43-s − 1.10e7·45-s − 3.09e7·47-s − 1.17e7·49-s − 3.01e7·51-s + 1.00e8·53-s + 1.01e7·55-s + ⋯
L(s)  = 1  + 0.655·3-s + 0.894·5-s + 1.08·7-s − 0.448·9-s + 0.166·11-s + 1.08·13-s + 0.586·15-s − 0.951·17-s + 2.03·19-s + 0.713·21-s + 0.787·23-s + 3/5·25-s − 0.214·27-s − 1.10·29-s + 2.20·31-s + 0.109·33-s + 0.972·35-s − 0.636·37-s + 0.712·39-s + 0.720·41-s + 2.13·43-s − 0.400·45-s − 0.924·47-s − 0.292·49-s − 0.623·51-s + 1.75·53-s + 0.148·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1697.67\)
Root analytic conductor: \(6.41894\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6400,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(7.469458423\)
\(L(\frac12)\) \(\approx\) \(7.469458423\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 - p^{4} T )^{2} \)
good3$D_{4}$ \( 1 - 92 T + 5762 p T^{2} - 92 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 - 6908 T + 8501858 p T^{2} - 6908 p^{9} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 - 8080 T + 4065472006 T^{2} - 8080 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 111948 T + 22805544638 T^{2} - 111948 p^{9} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 + 327532 T + 241881460294 T^{2} + 327532 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 - 1156680 T + 686155308982 T^{2} - 1156680 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 - 1057252 T + 1690239470158 T^{2} - 1057252 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 4212260 T + 32604383130814 T^{2} + 4212260 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 - 366488 p T + 71055092544062 T^{2} - 366488 p^{10} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 + 7251860 T + 207906306187118 T^{2} + 7251860 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 13030436 T - 112698304084010 T^{2} - 13030436 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 - 47934076 T + 1577772318092534 T^{2} - 47934076 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 + 30914292 T + 2034610190905950 T^{2} + 30914292 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 - 100922108 T + 9120851179993582 T^{2} - 100922108 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 - 47362536 T + 6051611540480326 T^{2} - 47362536 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 203634428 T + 30920737906297022 T^{2} - 203634428 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 - 58872852 T + 54767076047787046 T^{2} - 58872852 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 - 349900792 T + 118671349360183822 T^{2} - 349900792 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 - 71160388 T - 50883178339169962 T^{2} - 71160388 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 - 452087344 T + 272781342616725918 T^{2} - 452087344 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 - 244865436 T + 172811505943449574 T^{2} - 244865436 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 + 256073260 T + 592174274852066582 T^{2} + 256073260 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 + 184950572 T - 615454564650127770 T^{2} + 184950572 p^{9} T^{3} + p^{18} T^{4} \)
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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

−13.03648815436483726393890411857, −12.23421613119468086224242107921, −11.38273765872708995286430923241, −11.38065384647621507750883692489, −10.65855012532248565811235559117, −9.952453989388518201095623764069, −9.141559742190333412587278347040, −9.120467848652143624364575188988, −8.144720079828455415709572703468, −8.015251403483232339354963258618, −6.96647734586273725142928649162, −6.48600773962170908525759908493, −5.42770185855428129866243821939, −5.35276953655564199319755521546, −4.31557160288311971076449864971, −3.55196807593875496463169321211, −2.68546669777463419980060802940, −2.21093283835577647374650659519, −1.19215639251239722226521382663, −0.873161564069951135021334074971, 0.873161564069951135021334074971, 1.19215639251239722226521382663, 2.21093283835577647374650659519, 2.68546669777463419980060802940, 3.55196807593875496463169321211, 4.31557160288311971076449864971, 5.35276953655564199319755521546, 5.42770185855428129866243821939, 6.48600773962170908525759908493, 6.96647734586273725142928649162, 8.015251403483232339354963258618, 8.144720079828455415709572703468, 9.120467848652143624364575188988, 9.141559742190333412587278347040, 9.952453989388518201095623764069, 10.65855012532248565811235559117, 11.38065384647621507750883692489, 11.38273765872708995286430923241, 12.23421613119468086224242107921, 13.03648815436483726393890411857

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