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Properties

Label 8-98e4-1.1-c9e4-0-2
Degree $8$
Conductor $92236816$
Sign $1$
Analytic cond. $6.49014\times 10^{6}$
Root an. cond. $7.10447$
Motivic weight $9$
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  + 32·2-s − 14·3-s + 256·4-s − 2.73e3·5-s − 448·6-s − 8.19e3·8-s − 1.82e4·9-s − 8.73e4·10-s − 4.49e4·11-s − 3.58e3·12-s − 2.00e5·13-s + 3.82e4·15-s − 2.62e5·16-s − 8.70e5·17-s − 5.82e5·18-s + 5.08e5·19-s − 6.98e5·20-s − 1.43e6·22-s − 7.98e4·23-s + 1.14e5·24-s + 4.75e6·25-s − 6.41e6·26-s + 1.31e6·27-s + 4.01e6·29-s + 1.22e6·30-s + 2.18e6·31-s − 2.09e6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.0997·3-s + 1/2·4-s − 1.95·5-s − 0.141·6-s − 0.707·8-s − 0.925·9-s − 2.76·10-s − 0.925·11-s − 0.0498·12-s − 1.94·13-s + 0.194·15-s − 16-s − 2.52·17-s − 1.30·18-s + 0.895·19-s − 0.976·20-s − 1.30·22-s − 0.0594·23-s + 0.0705·24-s + 2.43·25-s − 2.75·26-s + 0.477·27-s + 1.05·29-s + 0.275·30-s + 0.425·31-s − 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(92236816\)    =    \(2^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.49014\times 10^{6}\)
Root analytic conductor: \(7.10447\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 92236816,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.721709812\)
\(L(\frac12)\) \(\approx\) \(1.721709812\)
\(L(\frac{11}{2})\) 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$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 14 T + 18406 T^{2} - 268688 p T^{3} - 6630473 p^{2} T^{4} - 268688 p^{10} T^{5} + 18406 p^{18} T^{6} + 14 p^{27} T^{7} + p^{36} T^{8} \)
5$D_4\times C_2$ \( 1 + 546 p T + 539986 p T^{2} + 92461824 p^{2} T^{3} + 168783249711 p^{2} T^{4} + 92461824 p^{11} T^{5} + 539986 p^{19} T^{6} + 546 p^{28} T^{7} + p^{36} T^{8} \)
11$D_4\times C_2$ \( 1 + 44940 T - 659930182 T^{2} - 91514090304000 T^{3} - 3142135669968295557 T^{4} - 91514090304000 p^{9} T^{5} - 659930182 p^{18} T^{6} + 44940 p^{27} T^{7} + p^{36} T^{8} \)
13$D_{4}$ \( ( 1 + 7714 p T + 23606428002 T^{2} + 7714 p^{10} T^{3} + p^{18} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 870408 T + 336859082354 T^{2} + 159785367173375328 T^{3} + \)\(73\!\cdots\!83\)\( T^{4} + 159785367173375328 p^{9} T^{5} + 336859082354 p^{18} T^{6} + 870408 p^{27} T^{7} + p^{36} T^{8} \)
19$D_4\times C_2$ \( 1 - 508774 T - 33946815626 T^{2} + 9441174434884976 p T^{3} - \)\(28\!\cdots\!81\)\( p^{2} T^{4} + 9441174434884976 p^{10} T^{5} - 33946815626 p^{18} T^{6} - 508774 p^{27} T^{7} + p^{36} T^{8} \)
23$D_4\times C_2$ \( 1 + 79800 T - 151305274562 p T^{2} - 9250094246400000 T^{3} + \)\(88\!\cdots\!07\)\( T^{4} - 9250094246400000 p^{9} T^{5} - 151305274562 p^{19} T^{6} + 79800 p^{27} T^{7} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 - 2006328 T - 2889837567866 T^{2} - 2006328 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 2188732 T - 45720846780974 T^{2} + 5182588568359774208 T^{3} + \)\(17\!\cdots\!59\)\( T^{4} + 5182588568359774208 p^{9} T^{5} - 45720846780974 p^{18} T^{6} - 2188732 p^{27} T^{7} + p^{36} T^{8} \)
37$D_4\times C_2$ \( 1 - 20723576 T + 109318009611178 T^{2} - \)\(12\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!44\)\( p^{9} T^{5} + 109318009611178 p^{18} T^{6} - 20723576 p^{27} T^{7} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 19016592 T + 126947391521038 T^{2} + 19016592 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 4193716 T + 733843976191350 T^{2} - 4193716 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 74542524 T + 2021776853206898 T^{2} + \)\(96\!\cdots\!56\)\( T^{3} + \)\(50\!\cdots\!03\)\( T^{4} + \)\(96\!\cdots\!56\)\( p^{9} T^{5} + 2021776853206898 p^{18} T^{6} + 74542524 p^{27} T^{7} + p^{36} T^{8} \)
53$D_4\times C_2$ \( 1 - 3239748 T - 6094285004204638 T^{2} + \)\(16\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!43\)\( T^{4} + \)\(16\!\cdots\!52\)\( p^{9} T^{5} - 6094285004204638 p^{18} T^{6} - 3239748 p^{27} T^{7} + p^{36} T^{8} \)
59$D_4\times C_2$ \( 1 + 133642362 T - 1040757032736970 T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!59\)\( T^{4} + \)\(21\!\cdots\!32\)\( p^{9} T^{5} - 1040757032736970 p^{18} T^{6} + 133642362 p^{27} T^{7} + p^{36} T^{8} \)
61$D_4\times C_2$ \( 1 - 227801686 T + 17261381012831290 T^{2} - \)\(25\!\cdots\!64\)\( T^{3} + \)\(45\!\cdots\!19\)\( T^{4} - \)\(25\!\cdots\!64\)\( p^{9} T^{5} + 17261381012831290 p^{18} T^{6} - 227801686 p^{27} T^{7} + p^{36} T^{8} \)
67$D_4\times C_2$ \( 1 + 332930272 T + 35027578907947594 T^{2} + \)\(71\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!43\)\( T^{4} + \)\(71\!\cdots\!12\)\( p^{9} T^{5} + 35027578907947594 p^{18} T^{6} + 332930272 p^{27} T^{7} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 + 167985720 T + 6569741497979662 T^{2} + 167985720 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 44684276 T - 32068117882924694 T^{2} - \)\(37\!\cdots\!56\)\( T^{3} - \)\(24\!\cdots\!57\)\( T^{4} - \)\(37\!\cdots\!56\)\( p^{9} T^{5} - 32068117882924694 p^{18} T^{6} + 44684276 p^{27} T^{7} + p^{36} T^{8} \)
79$D_4\times C_2$ \( 1 + 269642776 T - 173875969301015006 T^{2} + \)\(18\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!19\)\( T^{4} + \)\(18\!\cdots\!44\)\( p^{9} T^{5} - 173875969301015006 p^{18} T^{6} + 269642776 p^{27} T^{7} + p^{36} T^{8} \)
83$D_{4}$ \( ( 1 - 183105762 T + 297182791992067342 T^{2} - 183105762 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 791657748 T - 124556565155958790 T^{2} - \)\(40\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!19\)\( T^{4} - \)\(40\!\cdots\!48\)\( p^{9} T^{5} - 124556565155958790 p^{18} T^{6} - 791657748 p^{27} T^{7} + p^{36} T^{8} \)
97$D_{4}$ \( ( 1 - 4169480 T + 1069351625837487534 T^{2} - 4169480 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
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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

−8.307939008907391870415965492917, −8.071514478501575026430635160510, −7.79566653066327809457792783334, −7.56093996004129760527915242047, −7.07201734031551310836812044578, −6.94415248177636022159732126475, −6.48186280697066511014679406016, −6.42238613589100715165077191743, −5.85444533645914575504784055939, −5.48762747301293078110482773328, −5.11059738663777531974530307820, −4.80647857978046860758852635187, −4.58220704161097518517820239335, −4.53458336564717523425013293947, −4.22873637516757481775355705676, −3.63795100770105914441318346203, −3.24724926992915283808986015837, −3.04619584881780318596245854883, −2.62542970928684514826467757201, −2.61782400495027146975602945582, −1.96222437478047089739850809669, −1.51504807530243233297711113916, −0.59015703282698724450110333784, −0.43003862669511738283777747040, −0.29918598244188736583183443058, 0.29918598244188736583183443058, 0.43003862669511738283777747040, 0.59015703282698724450110333784, 1.51504807530243233297711113916, 1.96222437478047089739850809669, 2.61782400495027146975602945582, 2.62542970928684514826467757201, 3.04619584881780318596245854883, 3.24724926992915283808986015837, 3.63795100770105914441318346203, 4.22873637516757481775355705676, 4.53458336564717523425013293947, 4.58220704161097518517820239335, 4.80647857978046860758852635187, 5.11059738663777531974530307820, 5.48762747301293078110482773328, 5.85444533645914575504784055939, 6.42238613589100715165077191743, 6.48186280697066511014679406016, 6.94415248177636022159732126475, 7.07201734031551310836812044578, 7.56093996004129760527915242047, 7.79566653066327809457792783334, 8.071514478501575026430635160510, 8.307939008907391870415965492917

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