Properties

Label 8-2520e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.033\times 10^{13}$
Sign $1$
Analytic cond. $163949.$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s − 8·19-s + 10·25-s − 16·29-s + 16·31-s + 32·41-s − 2·49-s − 32·61-s − 32·71-s − 16·89-s − 24·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2.41·11-s − 1.83·19-s + 2·25-s − 2.97·29-s + 2.87·31-s + 4.99·41-s − 2/7·49-s − 4.09·61-s − 3.79·71-s − 1.69·89-s − 2.29·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(163949.\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1836693796\)
\(L(\frac12)\) \(\approx\) \(0.1836693796\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_4$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
41$C_4$ \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_4$ \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.32435016968409254855350217916, −6.23023482819972037471261547844, −6.10853163606682044385649771248, −5.92277524434864579312820268897, −5.44393175134077289454331760122, −5.29541849731948431055297063946, −5.14515283166051883873038889939, −4.74989833386195693232903285521, −4.43424640636858183694546440247, −4.37135113917201582958858077952, −4.21733071223587649787686403617, −4.11062186266416378142198170852, −3.86584619562985714261504893541, −3.75491469732468528121836323025, −3.02848833611702520020621631106, −3.02688131945124668336337135387, −2.82512604558852880318999485056, −2.74685062295910314852673802033, −2.16082444557884008439813359293, −2.09964353407427040330629333812, −1.52329529041637748543082846219, −1.40149575098521119209871460294, −1.04021977116270304535962671624, −1.01294888037099936433042195482, −0.06317923134618627038059365220, 0.06317923134618627038059365220, 1.01294888037099936433042195482, 1.04021977116270304535962671624, 1.40149575098521119209871460294, 1.52329529041637748543082846219, 2.09964353407427040330629333812, 2.16082444557884008439813359293, 2.74685062295910314852673802033, 2.82512604558852880318999485056, 3.02688131945124668336337135387, 3.02848833611702520020621631106, 3.75491469732468528121836323025, 3.86584619562985714261504893541, 4.11062186266416378142198170852, 4.21733071223587649787686403617, 4.37135113917201582958858077952, 4.43424640636858183694546440247, 4.74989833386195693232903285521, 5.14515283166051883873038889939, 5.29541849731948431055297063946, 5.44393175134077289454331760122, 5.92277524434864579312820268897, 6.10853163606682044385649771248, 6.23023482819972037471261547844, 6.32435016968409254855350217916

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