Properties

Label 8-1950e4-1.1-c1e4-0-25
Degree $8$
Conductor $1.446\times 10^{13}$
Sign $1$
Analytic cond. $58782.3$
Root an. cond. $3.94598$
Motivic weight $1$
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-s + 9-s − 2·11-s + 12·19-s − 2·29-s − 12·31-s + 36-s − 20·41-s − 2·44-s − 10·49-s − 10·59-s + 20·61-s − 64-s − 8·71-s + 12·76-s − 20·79-s + 20·89-s − 2·99-s − 28·101-s + 24·109-s − 2·116-s + 23·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 0.603·11-s + 2.75·19-s − 0.371·29-s − 2.15·31-s + 1/6·36-s − 3.12·41-s − 0.301·44-s − 1.42·49-s − 1.30·59-s + 2.56·61-s − 1/8·64-s − 0.949·71-s + 1.37·76-s − 2.25·79-s + 2.11·89-s − 0.201·99-s − 2.78·101-s + 2.29·109-s − 0.185·116-s + 2.09·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58782.3\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.309311361\)
\(L(\frac12)\) \(\approx\) \(3.309311361\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
good7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \)
47$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.76484522833525095049920523932, −6.29057275290552530694378593564, −6.09323311069289885708756452100, −5.67828864075105695272961870797, −5.64013268809588417002840465649, −5.55155475430774010101907695536, −5.32341725662420127700357347207, −5.24300851359904755793611236983, −4.76336939744102833938520561308, −4.50163313356322319084227460886, −4.48180440261855053376461244759, −4.33018057398361722521407923271, −3.70540931545963641394032382478, −3.55746447600528385135268560679, −3.34708447421399459325475556642, −3.13073454544580822823201842312, −2.99192935752811538008771822665, −2.98229312493559604269332593059, −2.11057633957940645040795954378, −2.05561012160042341254413756825, −1.82319520435360529004183313402, −1.66406921390254357522993815520, −1.19080951470495615435194750177, −0.71352160374702783716828673267, −0.35947495459959203342252032191, 0.35947495459959203342252032191, 0.71352160374702783716828673267, 1.19080951470495615435194750177, 1.66406921390254357522993815520, 1.82319520435360529004183313402, 2.05561012160042341254413756825, 2.11057633957940645040795954378, 2.98229312493559604269332593059, 2.99192935752811538008771822665, 3.13073454544580822823201842312, 3.34708447421399459325475556642, 3.55746447600528385135268560679, 3.70540931545963641394032382478, 4.33018057398361722521407923271, 4.48180440261855053376461244759, 4.50163313356322319084227460886, 4.76336939744102833938520561308, 5.24300851359904755793611236983, 5.32341725662420127700357347207, 5.55155475430774010101907695536, 5.64013268809588417002840465649, 5.67828864075105695272961870797, 6.09323311069289885708756452100, 6.29057275290552530694378593564, 6.76484522833525095049920523932

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