Properties

Label 8-882e4-1.1-c3e4-0-7
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $7.33396\times 10^{6}$
Root an. cond. $7.21385$
Motivic weight $3$
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  − 4·2-s + 4·4-s + 16·8-s − 28·11-s − 64·16-s + 112·22-s + 280·23-s − 142·25-s + 1.14e3·29-s + 64·32-s + 76·37-s − 136·43-s − 112·44-s − 1.12e3·46-s + 568·50-s − 148·53-s − 4.57e3·58-s + 192·64-s − 1.36e3·67-s − 2.35e3·71-s − 304·74-s − 2.44e3·79-s + 544·86-s − 448·88-s + 1.12e3·92-s − 568·100-s + 592·106-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.707·8-s − 0.767·11-s − 16-s + 1.08·22-s + 2.53·23-s − 1.13·25-s + 7.32·29-s + 0.353·32-s + 0.337·37-s − 0.482·43-s − 0.383·44-s − 3.58·46-s + 1.60·50-s − 0.383·53-s − 10.3·58-s + 3/8·64-s − 2.49·67-s − 3.93·71-s − 0.477·74-s − 3.47·79-s + 0.682·86-s − 0.542·88-s + 1.26·92-s − 0.567·100-s + 0.542·106-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(7.33396\times 10^{6}\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.958764154\)
\(L(\frac12)\) \(\approx\) \(1.958764154\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 233 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 233 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} ) \)
11$C_2^2$ \( ( 1 + 14 T - 1135 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 1802 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^3$ \( 1 - 13716 T^{2} + 141082775 T^{4} - 13716 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 - 140 T + 7433 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 286 T + p^{3} T^{2} )^{4} \)
31$C_2^3$ \( 1 - 50870 T^{2} + 1700253219 T^{4} - 50870 p^{6} T^{6} + p^{12} T^{8} \)
37$C_2^2$ \( ( 1 - 38 T - 49209 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 122000 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{4} \)
47$C_2^3$ \( 1 + 66154 T^{2} - 6402863613 T^{4} + 66154 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^2$ \( ( 1 + 74 T - 143401 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 222260 T^{2} + 7218973959 T^{4} - 222260 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^3$ \( 1 - 453762 T^{2} + 154379578283 T^{4} - 453762 p^{6} T^{6} + p^{12} T^{8} \)
67$C_2^2$ \( ( 1 + 684 T + 167093 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 588 T + p^{3} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 705072 T^{2} + 345792298895 T^{4} - 705072 p^{6} T^{6} + p^{12} T^{8} \)
79$C_2^2$ \( ( 1 + 1220 T + 995361 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 964772 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 1028000 T^{2} + 559802709039 T^{4} - 1028000 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^2$ \( ( 1 - 375456 T^{2} + p^{6} T^{4} )^{2} \)
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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

−6.92108630286864263786261003014, −6.76550200499265816108819115919, −6.65514580025969549666061154320, −5.97519078801173983788063913493, −5.92602239707001080931848776923, −5.92221404011667504073193831164, −5.77116766179891848188545536818, −4.98211292346694978585814043897, −4.79747561396051656424404861612, −4.71090521018591599807868959071, −4.57921384408941327218223681188, −4.55201335185309280783834332753, −4.17333641762579906760814099454, −3.53652558586709856623007971661, −3.23853276184909593166650410231, −3.09464924440503464844772903846, −2.63256124699077910833890179496, −2.63019114901879984600017364355, −2.61534311121478685466171515051, −1.60287146094368946268451235143, −1.46293490880201235149352338711, −1.34860642810361518569607493128, −0.74287144411068182297460020405, −0.68548417447671785441001962924, −0.29449209987734126769555513327, 0.29449209987734126769555513327, 0.68548417447671785441001962924, 0.74287144411068182297460020405, 1.34860642810361518569607493128, 1.46293490880201235149352338711, 1.60287146094368946268451235143, 2.61534311121478685466171515051, 2.63019114901879984600017364355, 2.63256124699077910833890179496, 3.09464924440503464844772903846, 3.23853276184909593166650410231, 3.53652558586709856623007971661, 4.17333641762579906760814099454, 4.55201335185309280783834332753, 4.57921384408941327218223681188, 4.71090521018591599807868959071, 4.79747561396051656424404861612, 4.98211292346694978585814043897, 5.77116766179891848188545536818, 5.92221404011667504073193831164, 5.92602239707001080931848776923, 5.97519078801173983788063913493, 6.65514580025969549666061154320, 6.76550200499265816108819115919, 6.92108630286864263786261003014

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