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Properties

Label 8-560e4-1.1-c3e4-0-3
Degree $8$
Conductor $98344960000$
Sign $1$
Analytic cond. $1.19183\times 10^{6}$
Root an. cond. $5.74813$
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  − 2·3-s − 10·5-s − 6·7-s + 9·9-s − 68·11-s − 104·13-s + 20·15-s − 164·17-s − 232·19-s + 12·21-s − 198·23-s + 25·25-s + 62·27-s − 36·29-s + 196·31-s + 136·33-s + 60·35-s − 160·37-s + 208·39-s + 124·41-s − 396·43-s − 90·45-s + 164·47-s + 491·49-s + 328·51-s − 40·53-s + 680·55-s + ⋯
L(s)  = 1  − 0.384·3-s − 0.894·5-s − 0.323·7-s + 1/3·9-s − 1.86·11-s − 2.21·13-s + 0.344·15-s − 2.33·17-s − 2.80·19-s + 0.124·21-s − 1.79·23-s + 1/5·25-s + 0.441·27-s − 0.230·29-s + 1.13·31-s + 0.717·33-s + 0.289·35-s − 0.710·37-s + 0.854·39-s + 0.472·41-s − 1.40·43-s − 0.298·45-s + 0.508·47-s + 1.43·49-s + 0.900·51-s − 0.103·53-s + 1.66·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.19183\times 10^{6}\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1700846650\)
\(L(\frac12)\) \(\approx\) \(0.1700846650\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T - 65 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
good3$D_4\times C_2$ \( 1 + 2 T - 5 T^{2} - 10 p^{2} T^{3} - 28 p^{3} T^{4} - 10 p^{5} T^{5} - 5 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 68 T + 90 p T^{2} + 66096 T^{3} + 5279851 T^{4} + 66096 p^{3} T^{5} + 90 p^{7} T^{6} + 68 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 + 4 p T + 4334 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 164 T + 13290 T^{2} + 619920 T^{3} + 28845619 T^{4} + 619920 p^{3} T^{5} + 13290 p^{6} T^{6} + 164 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 232 T + 27386 T^{2} + 2951040 T^{3} + 282743147 T^{4} + 2951040 p^{3} T^{5} + 27386 p^{6} T^{6} + 232 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 198 T + 15419 T^{2} - 108702 T^{3} - 59304732 T^{4} - 108702 p^{3} T^{5} + 15419 p^{6} T^{6} + 198 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 18 T + 33955 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 196 T + 10630 T^{2} + 6232016 T^{3} - 1259892941 T^{4} + 6232016 p^{3} T^{5} + 10630 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 160 T - 75482 T^{2} - 35840 T^{3} + 6355127515 T^{4} - 35840 p^{3} T^{5} - 75482 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 62 T + 72379 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 198 T + 140065 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 164 T - 184530 T^{2} - 619920 T^{3} + 31129314739 T^{4} - 619920 p^{3} T^{5} - 184530 p^{6} T^{6} - 164 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 40 T - 293610 T^{2} - 1920 p T^{3} + 23049619 p^{2} T^{4} - 1920 p^{4} T^{5} - 293610 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 80 T + 91578 T^{2} - 39674880 T^{3} - 36239667557 T^{4} - 39674880 p^{3} T^{5} + 91578 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 174 T - 75031 T^{2} + 60665970 T^{3} - 47302886868 T^{4} + 60665970 p^{3} T^{5} - 75031 p^{6} T^{6} - 174 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 1054 T + 438155 T^{2} + 75081690 T^{3} + 28026552044 T^{4} + 75081690 p^{3} T^{5} + 438155 p^{6} T^{6} + 1054 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 832 T + 815278 T^{2} - 832 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 820 T - 167750 T^{2} + 50935120 T^{3} + 292942765411 T^{4} + 50935120 p^{3} T^{5} - 167750 p^{6} T^{6} + 820 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 576 T - 548830 T^{2} - 60751872 T^{3} + 368800964451 T^{4} - 60751872 p^{3} T^{5} - 548830 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 298 T + 423841 T^{2} - 298 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 182 T - 1119399 T^{2} + 46849530 T^{3} + 807976581412 T^{4} + 46849530 p^{3} T^{5} - 1119399 p^{6} T^{6} - 182 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 892 T + 1918278 T^{2} - 892 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−7.49840172839846975920152725970, −7.22204896203011736668384343764, −6.70228473640366568453093137206, −6.68687580040859031338252619385, −6.38002509261030829542760522809, −6.16239475704585380436550105202, −6.08924975963076299865793874702, −5.59235313013066754470273324314, −5.11768871470364403139887960151, −5.06708272977740828474948390732, −4.94503994509212419873705115272, −4.42210546374807196448679986086, −4.40618536095827879252028623902, −4.11172463549694587604447638445, −4.00512812171798283210801282055, −3.44491931409371985000113464396, −3.12689606628372631191637497092, −2.60748184119925085137356596372, −2.38518499868914868171127811252, −2.37706643022658589229465719963, −1.92478450766304125532104364881, −1.77257777029444443664206092028, −0.72312432879610718062470151297, −0.33699206853190655328079175809, −0.15569580553852067243695184662, 0.15569580553852067243695184662, 0.33699206853190655328079175809, 0.72312432879610718062470151297, 1.77257777029444443664206092028, 1.92478450766304125532104364881, 2.37706643022658589229465719963, 2.38518499868914868171127811252, 2.60748184119925085137356596372, 3.12689606628372631191637497092, 3.44491931409371985000113464396, 4.00512812171798283210801282055, 4.11172463549694587604447638445, 4.40618536095827879252028623902, 4.42210546374807196448679986086, 4.94503994509212419873705115272, 5.06708272977740828474948390732, 5.11768871470364403139887960151, 5.59235313013066754470273324314, 6.08924975963076299865793874702, 6.16239475704585380436550105202, 6.38002509261030829542760522809, 6.68687580040859031338252619385, 6.70228473640366568453093137206, 7.22204896203011736668384343764, 7.49840172839846975920152725970

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