Properties

Label 8-1232e4-1.1-c1e4-0-2
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $9365.93$
Root an. cond. $3.13649$
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  − 2·5-s − 9-s + 2·11-s + 20·13-s − 12·17-s − 6·19-s − 2·23-s + 4·25-s + 4·29-s − 8·31-s − 2·37-s + 12·41-s + 16·43-s + 2·45-s + 16·47-s − 7·49-s + 2·53-s − 4·55-s + 4·59-s − 18·61-s − 40·65-s − 8·67-s − 28·71-s − 6·73-s + 9·81-s − 32·83-s + 24·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s + 0.603·11-s + 5.54·13-s − 2.91·17-s − 1.37·19-s − 0.417·23-s + 4/5·25-s + 0.742·29-s − 1.43·31-s − 0.328·37-s + 1.87·41-s + 2.43·43-s + 0.298·45-s + 2.33·47-s − 49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 2.30·61-s − 4.96·65-s − 0.977·67-s − 3.32·71-s − 0.702·73-s + 81-s − 3.51·83-s + 2.60·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(9365.93\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5230401568\)
\(L(\frac12)\) \(\approx\) \(0.5230401568\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good3$C_2^3$ \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 6 T - 4 T^{2} + 12 T^{3} + 555 T^{4} + 12 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$D_4\times C_2$ \( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 - 16 T + 126 T^{2} - 576 T^{3} + 2659 T^{4} - 576 p T^{5} + 126 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 4 T - 99 T^{2} + 12 T^{3} + 8800 T^{4} + 12 p T^{5} - 99 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 8 T - 23 T^{2} - 376 T^{3} - 1208 T^{4} - 376 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 768 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \)
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

−7.06815205032687798235384504273, −6.45594071446622869540642642156, −6.39806735423414835507748121802, −6.37592617205597800754875725509, −6.14776889474822562283330918514, −5.87966854366577801474212495871, −5.73312593564864172002434468158, −5.63040366250997921273855025475, −5.27862558072729570742165936858, −4.69593950748203889698440332178, −4.36104354397071741211538853652, −4.34861408024812125919275887923, −4.03249708460195424156216601753, −3.98123321716685295986276572219, −3.97445699120429548594873416099, −3.66002031534874213529928011374, −3.02041335989670202357042529529, −2.95769188044307850401737459651, −2.74630877004431434470935594187, −2.28168011392371954220428266392, −1.92763117877507553264899257053, −1.33300736076809306614898087650, −1.27106890409501366008179539481, −1.16351444443554253146833506058, −0.15241677916266985971503274949, 0.15241677916266985971503274949, 1.16351444443554253146833506058, 1.27106890409501366008179539481, 1.33300736076809306614898087650, 1.92763117877507553264899257053, 2.28168011392371954220428266392, 2.74630877004431434470935594187, 2.95769188044307850401737459651, 3.02041335989670202357042529529, 3.66002031534874213529928011374, 3.97445699120429548594873416099, 3.98123321716685295986276572219, 4.03249708460195424156216601753, 4.34861408024812125919275887923, 4.36104354397071741211538853652, 4.69593950748203889698440332178, 5.27862558072729570742165936858, 5.63040366250997921273855025475, 5.73312593564864172002434468158, 5.87966854366577801474212495871, 6.14776889474822562283330918514, 6.37592617205597800754875725509, 6.39806735423414835507748121802, 6.45594071446622869540642642156, 7.06815205032687798235384504273

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