Properties

Label 8-980e4-1.1-c1e4-0-16
Degree $8$
Conductor $922368160000$
Sign $1$
Analytic cond. $3749.83$
Root an. cond. $2.79738$
Motivic weight $1$
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  + 2·2-s + 2·4-s − 4·5-s + 4·8-s − 8·10-s + 4·13-s + 8·16-s + 6·17-s − 8·20-s + 5·25-s + 8·26-s + 8·32-s + 12·34-s + 14·37-s − 16·40-s + 32·41-s + 10·50-s + 8·52-s − 18·53-s + 24·61-s + 8·64-s − 16·65-s + 12·68-s − 22·73-s + 28·74-s − 32·80-s + 9·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s + 1.10·13-s + 2·16-s + 1.45·17-s − 1.78·20-s + 25-s + 1.56·26-s + 1.41·32-s + 2.05·34-s + 2.30·37-s − 2.52·40-s + 4.99·41-s + 1.41·50-s + 1.10·52-s − 2.47·53-s + 3.07·61-s + 64-s − 1.98·65-s + 1.45·68-s − 2.57·73-s + 3.25·74-s − 3.57·80-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3749.83\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.902521712\)
\(L(\frac12)\) \(\approx\) \(6.902521712\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
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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

−7.32348334146606884167499512863, −7.06803307519818886681158591176, −6.48622191686268831445444489127, −6.42778427606911589056697556357, −6.35294445817611710197647100692, −5.84560053950339525983148681579, −5.77163018534727441130035604470, −5.57484291456888594466816570042, −5.25732030553520661090493643061, −5.21702369564741684539490114235, −4.56876047824753049778130665545, −4.50708507308186930885239646859, −4.27463355152783802809991195544, −4.09496297284931900555218036566, −3.95104313842934157132971414662, −3.77957750497708541029536593276, −3.41026813961926222205522704825, −3.11798097105132635371657819136, −2.74645250731402683034128553097, −2.68400506719094407886163835143, −2.30196820499498107034285342555, −1.64709144584983628216749916858, −1.25328371338526509910136097229, −1.04080545308933562506111898808, −0.54888306258959967741179582848, 0.54888306258959967741179582848, 1.04080545308933562506111898808, 1.25328371338526509910136097229, 1.64709144584983628216749916858, 2.30196820499498107034285342555, 2.68400506719094407886163835143, 2.74645250731402683034128553097, 3.11798097105132635371657819136, 3.41026813961926222205522704825, 3.77957750497708541029536593276, 3.95104313842934157132971414662, 4.09496297284931900555218036566, 4.27463355152783802809991195544, 4.50708507308186930885239646859, 4.56876047824753049778130665545, 5.21702369564741684539490114235, 5.25732030553520661090493643061, 5.57484291456888594466816570042, 5.77163018534727441130035604470, 5.84560053950339525983148681579, 6.35294445817611710197647100692, 6.42778427606911589056697556357, 6.48622191686268831445444489127, 7.06803307519818886681158591176, 7.32348334146606884167499512863

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