Properties

Label 8-1014e4-1.1-c2e4-0-5
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $582763.$
Root an. cond. $5.25637$
Motivic weight $2$
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  − 4·2-s + 8·4-s + 6·5-s + 8·7-s − 8·8-s + 6·9-s − 24·10-s + 24·11-s − 32·14-s − 4·16-s − 24·18-s + 8·19-s + 48·20-s − 96·22-s + 18·25-s + 64·28-s + 120·29-s + 88·31-s + 32·32-s + 48·35-s + 48·36-s − 26·37-s − 32·38-s − 48·40-s + 90·41-s + 192·44-s + 36·45-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s + 6/5·5-s + 8/7·7-s − 8-s + 2/3·9-s − 2.39·10-s + 2.18·11-s − 2.28·14-s − 1/4·16-s − 4/3·18-s + 8/19·19-s + 12/5·20-s − 4.36·22-s + 0.719·25-s + 16/7·28-s + 4.13·29-s + 2.83·31-s + 32-s + 1.37·35-s + 4/3·36-s − 0.702·37-s − 0.842·38-s − 6/5·40-s + 2.19·41-s + 4.36·44-s + 4/5·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(582763.\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.562055058\)
\(L(\frac12)\) \(\approx\) \(5.562055058\)
\(L(2)\) 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$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
13 \( 1 \)
good5$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 168 T^{3} + 1559 T^{4} - 168 p^{2} T^{5} + 18 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} + 312 T^{3} - 4702 T^{4} + 312 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 18242 T^{4} - 2328 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 250 T^{2} + 161499 T^{4} - 250 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} + 120 T^{3} - 140926 T^{4} + 120 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1444 T^{2} + 1065414 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 60 T + 2507 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 88 T + 3872 T^{2} - 167640 T^{3} + 6366914 T^{4} - 167640 p^{2} T^{5} + 3872 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 + 26 T + 338 T^{2} + 36816 T^{3} + 4007903 T^{4} + 36816 p^{2} T^{5} + 338 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 90 T + 4050 T^{2} - 242280 T^{3} + 13471607 T^{4} - 242280 p^{2} T^{5} + 4050 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 - 2692 T^{2} + 4500390 T^{4} - 2692 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2$ \( ( 1 + 60 T + 1800 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 30 T + 4391 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 120 T + 7200 T^{2} + 147000 T^{3} - 2088286 T^{4} + 147000 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 126 T + 10979 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 32 T + 512 T^{2} - 85536 T^{3} + 10992002 T^{4} - 85536 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \)
71$C_2^3$ \( 1 + 28493474 T^{4} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 + 178 T + 15842 T^{2} + 1362768 T^{3} + 111813743 T^{4} + 1362768 p^{2} T^{5} + 15842 p^{4} T^{6} + 178 p^{6} T^{7} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 96 T + 13058 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 168 T + 14112 T^{2} - 717864 T^{3} + 29673602 T^{4} - 717864 p^{2} T^{5} + 14112 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 156 T + 12168 T^{2} + 1257204 T^{3} + 129875918 T^{4} + 1257204 p^{2} T^{5} + 12168 p^{4} T^{6} + 156 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 188 T + 17672 T^{2} + 564 T^{3} - 88472818 T^{4} + 564 p^{2} T^{5} + 17672 p^{4} T^{6} + 188 p^{6} T^{7} + p^{8} T^{8} \)
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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

−6.98076803389121669999363026323, −6.65483862922658871839649737476, −6.50488578010465533579977811038, −6.40404603346644683309524248582, −6.28123580270895305355665820124, −5.92911935301665263513646899308, −5.71760919707511403053360982142, −5.43395843786662301366754928822, −4.91884518690822890139418370537, −4.84931875941109015420262819664, −4.47413317178207700624588692911, −4.32572319352401377193012807461, −4.24626545281463675063639655981, −4.23973220161979525798645810491, −3.24747860463740556800829061189, −3.02636242900316027323870299580, −2.92207289854067201066749098567, −2.84471917232409510169732470222, −2.15519472014188045843150095987, −1.86313082026748887761256138416, −1.40580896826460111860211250963, −1.38870421374729744442353849081, −1.31356046416288530104307388663, −0.69200083178071340909438860283, −0.57671891369582159978232604570, 0.57671891369582159978232604570, 0.69200083178071340909438860283, 1.31356046416288530104307388663, 1.38870421374729744442353849081, 1.40580896826460111860211250963, 1.86313082026748887761256138416, 2.15519472014188045843150095987, 2.84471917232409510169732470222, 2.92207289854067201066749098567, 3.02636242900316027323870299580, 3.24747860463740556800829061189, 4.23973220161979525798645810491, 4.24626545281463675063639655981, 4.32572319352401377193012807461, 4.47413317178207700624588692911, 4.84931875941109015420262819664, 4.91884518690822890139418370537, 5.43395843786662301366754928822, 5.71760919707511403053360982142, 5.92911935301665263513646899308, 6.28123580270895305355665820124, 6.40404603346644683309524248582, 6.50488578010465533579977811038, 6.65483862922658871839649737476, 6.98076803389121669999363026323

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