Properties

Label 8-800e4-1.1-c3e4-0-2
Degree $8$
Conductor $409600000000$
Sign $1$
Analytic cond. $4.96391\times 10^{6}$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 28·9-s − 128·11-s + 416·19-s + 584·29-s − 352·31-s + 200·41-s + 140·49-s + 1.44e3·59-s − 2.53e3·61-s + 96·71-s + 64·79-s + 666·81-s + 1.56e3·89-s − 3.58e3·99-s + 3.03e3·101-s − 88·109-s + 6.64e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.64e3·169-s + ⋯
L(s)  = 1  + 1.03·9-s − 3.50·11-s + 5.02·19-s + 3.73·29-s − 2.03·31-s + 0.761·41-s + 0.408·49-s + 3.17·59-s − 5.32·61-s + 0.160·71-s + 0.0911·79-s + 0.913·81-s + 1.85·89-s − 3.63·99-s + 2.98·101-s − 0.0773·109-s + 4.99·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.56·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4.96391\times 10^{6}\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.301279544\)
\(L(\frac12)\) \(\approx\) \(2.301279544\)
\(L(\frac{5}{2})\) 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$D_4\times C_2$ \( 1 - 28 T^{2} + 118 T^{4} - 28 p^{6} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 20 p T^{2} + 201798 T^{4} - 20 p^{7} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 64 T + 2822 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 5644 T^{2} + 17396118 T^{4} - 5644 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 + 8004 T^{2} + 64069958 T^{4} + 8004 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 - 208 T + 22998 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 35660 T^{2} + 572163078 T^{4} - 35660 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 292 T + 63950 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 176 T + 66462 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 126956 T^{2} + 8382243318 T^{4} - 126956 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 100 T + 101942 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 181628 T^{2} + 16244858838 T^{4} - 181628 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 367980 T^{2} + 55092760358 T^{4} - 367980 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 517932 T^{2} + 110011702070 T^{4} - 517932 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 - 720 T + 530758 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 820252 T^{2} + 319773835638 T^{4} - 820252 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 48 T - 424178 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 423068 T^{2} + 42042925734 T^{4} - 423068 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 32 T + 276318 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 913884 T^{2} + 728309594678 T^{4} - 913884 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 - 780 T + 1506742 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2829692 T^{2} + 3610578653574 T^{4} - 2829692 p^{6} T^{6} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.37314786277208605665737489776, −6.63825209234495555643575804778, −6.52002323991450036728206318355, −6.41737400558386080576112512025, −5.81486426049067208348246208269, −5.73824940016413426021125710826, −5.33386120027143945065662179375, −5.31318330632661678948244534625, −5.26999839133578978740249070269, −4.81131885933074272211397677079, −4.63019809577796688695139003746, −4.58317122258723395453631335242, −4.09044447530969178106741864209, −3.65850588052178923320716778089, −3.24267642440340881647057025021, −3.18400492723543502928456591983, −3.02922532728584215102160202706, −2.77057463021715911296445539636, −2.43860534226772672584026343548, −2.01420103992417456513893182011, −1.78627956515893386160201594992, −1.05609852248250352841088460853, −0.911617435469885382713317166326, −0.895207497406132492883374657595, −0.18996234300081259661680457244, 0.18996234300081259661680457244, 0.895207497406132492883374657595, 0.911617435469885382713317166326, 1.05609852248250352841088460853, 1.78627956515893386160201594992, 2.01420103992417456513893182011, 2.43860534226772672584026343548, 2.77057463021715911296445539636, 3.02922532728584215102160202706, 3.18400492723543502928456591983, 3.24267642440340881647057025021, 3.65850588052178923320716778089, 4.09044447530969178106741864209, 4.58317122258723395453631335242, 4.63019809577796688695139003746, 4.81131885933074272211397677079, 5.26999839133578978740249070269, 5.31318330632661678948244534625, 5.33386120027143945065662179375, 5.73824940016413426021125710826, 5.81486426049067208348246208269, 6.41737400558386080576112512025, 6.52002323991450036728206318355, 6.63825209234495555643575804778, 7.37314786277208605665737489776

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