Properties

Label 8-462e4-1.1-c5e4-0-2
Degree $8$
Conductor $45558341136$
Sign $1$
Analytic cond. $3.01446\times 10^{7}$
Root an. cond. $8.60798$
Motivic weight $5$
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  + 16·2-s + 36·3-s + 160·4-s + 56·5-s + 576·6-s + 196·7-s + 1.28e3·8-s + 810·9-s + 896·10-s − 484·11-s + 5.76e3·12-s + 758·13-s + 3.13e3·14-s + 2.01e3·15-s + 8.96e3·16-s + 666·17-s + 1.29e4·18-s + 2.81e3·19-s + 8.96e3·20-s + 7.05e3·21-s − 7.74e3·22-s + 2.39e3·23-s + 4.60e4·24-s − 913·25-s + 1.21e4·26-s + 1.45e4·27-s + 3.13e4·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 1.00·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 10/3·9-s + 2.83·10-s − 1.20·11-s + 11.5·12-s + 1.24·13-s + 4.27·14-s + 2.31·15-s + 35/4·16-s + 0.558·17-s + 9.42·18-s + 1.78·19-s + 5.00·20-s + 3.49·21-s − 3.41·22-s + 0.942·23-s + 16.3·24-s − 0.292·25-s + 3.51·26-s + 3.84·27-s + 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.01446\times 10^{7}\)
Root analytic conductor: \(8.60798\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(605.9520623\)
\(L(\frac12)\) \(\approx\) \(605.9520623\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{4} \)
3$C_1$ \( ( 1 - p^{2} T )^{4} \)
7$C_1$ \( ( 1 - p^{2} T )^{4} \)
11$C_1$ \( ( 1 + p^{2} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 56 T + 4049 T^{2} - 137754 T^{3} + 340032 p T^{4} - 137754 p^{5} T^{5} + 4049 p^{10} T^{6} - 56 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 758 T + 1028401 T^{2} - 465212378 T^{3} + 427698246100 T^{4} - 465212378 p^{5} T^{5} + 1028401 p^{10} T^{6} - 758 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 666 T + 3146012 T^{2} - 2012855166 T^{3} + 382440646998 p T^{4} - 2012855166 p^{5} T^{5} + 3146012 p^{10} T^{6} - 666 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2816 T + 9718501 T^{2} - 15801249002 T^{3} + 33142150328596 T^{4} - 15801249002 p^{5} T^{5} + 9718501 p^{10} T^{6} - 2816 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 104 p T + 18568400 T^{2} - 50384926584 T^{3} + 155795481952446 T^{4} - 50384926584 p^{5} T^{5} + 18568400 p^{10} T^{6} - 104 p^{16} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4546 T + 72014033 T^{2} - 263159375190 T^{3} + 2128710975032292 T^{4} - 263159375190 p^{5} T^{5} + 72014033 p^{10} T^{6} - 4546 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 10586 T + 130914604 T^{2} - 825346469858 T^{3} + 5662370627936806 T^{4} - 825346469858 p^{5} T^{5} + 130914604 p^{10} T^{6} - 10586 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 11194 T + 5924149 p T^{2} - 2166636016958 T^{3} + 21386123777396420 T^{4} - 2166636016958 p^{5} T^{5} + 5924149 p^{11} T^{6} - 11194 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 6246 T + 260085116 T^{2} - 1293400002978 T^{3} + 36548745636880278 T^{4} - 1293400002978 p^{5} T^{5} + 260085116 p^{10} T^{6} - 6246 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 22392 T + 455013664 T^{2} - 6092318757464 T^{3} + 82358865286111950 T^{4} - 6092318757464 p^{5} T^{5} + 455013664 p^{10} T^{6} - 22392 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 1240 T + 636105317 T^{2} + 2176527093138 T^{3} + 189705123365008212 T^{4} + 2176527093138 p^{5} T^{5} + 636105317 p^{10} T^{6} + 1240 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 3352 T + 1497962372 T^{2} + 4368382680456 T^{3} + 903598414788254118 T^{4} + 4368382680456 p^{5} T^{5} + 1497962372 p^{10} T^{6} + 3352 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 14346 T + 1610691113 T^{2} + 19555337071062 T^{3} + 1291234446128070900 T^{4} + 19555337071062 p^{5} T^{5} + 1610691113 p^{10} T^{6} + 14346 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 16132 T + 1319934484 T^{2} - 15409127034764 T^{3} + 1052419714747164710 T^{4} - 15409127034764 p^{5} T^{5} + 1319934484 p^{10} T^{6} - 16132 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 14410 T + 4492100161 T^{2} - 36166466277362 T^{3} + 8357404344562643684 T^{4} - 36166466277362 p^{5} T^{5} + 4492100161 p^{10} T^{6} - 14410 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 67932 T + 7506666284 T^{2} - 327924334670796 T^{3} + 20316538016870520966 T^{4} - 327924334670796 p^{5} T^{5} + 7506666284 p^{10} T^{6} - 67932 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 3324 T + 1190167729 T^{2} + 18987632172538 T^{3} - 372447176582625072 T^{4} + 18987632172538 p^{5} T^{5} + 1190167729 p^{10} T^{6} - 3324 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 103344 T + 7856597260 T^{2} - 198577137682928 T^{3} + 9492808780804128678 T^{4} - 198577137682928 p^{5} T^{5} + 7856597260 p^{10} T^{6} - 103344 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 119562 T + 16816499012 T^{2} - 1234690798024746 T^{3} + 97571287397561124678 T^{4} - 1234690798024746 p^{5} T^{5} + 16816499012 p^{10} T^{6} - 119562 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 19680 T + 14596605980 T^{2} - 235955227701024 T^{3} + \)\(11\!\cdots\!78\)\( T^{4} - 235955227701024 p^{5} T^{5} + 14596605980 p^{10} T^{6} - 19680 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 32524 T + 31416715900 T^{2} - 781791334159508 T^{3} + \)\(39\!\cdots\!42\)\( T^{4} - 781791334159508 p^{5} T^{5} + 31416715900 p^{10} T^{6} - 32524 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.33216559632879062114481166493, −6.69398781065524483932865095223, −6.51670678881707777071023036623, −6.29861374789186723484136857063, −6.29694393117000423096146162170, −5.58601007354811792535216873481, −5.53923206590883713914504941286, −5.27290721488965225350095646853, −5.25102191208450114237796278366, −4.61279067847228392737524681052, −4.61241436224547603645837476159, −4.31532467388958677043156260569, −4.21430626985539337266046600980, −3.58611236898699830887034758017, −3.42879445479431456871175459132, −3.19059569794607831844606873987, −3.15602318452086993735784018124, −2.43462653396400279782313468720, −2.41011726472425407753287586624, −2.25111054772902646151763163268, −2.19061696542320993025704221137, −1.35677626324879243115710079327, −1.14185842458808843379654023110, −1.07654996973531929614227664566, −0.815093214019198522945764499033, 0.815093214019198522945764499033, 1.07654996973531929614227664566, 1.14185842458808843379654023110, 1.35677626324879243115710079327, 2.19061696542320993025704221137, 2.25111054772902646151763163268, 2.41011726472425407753287586624, 2.43462653396400279782313468720, 3.15602318452086993735784018124, 3.19059569794607831844606873987, 3.42879445479431456871175459132, 3.58611236898699830887034758017, 4.21430626985539337266046600980, 4.31532467388958677043156260569, 4.61241436224547603645837476159, 4.61279067847228392737524681052, 5.25102191208450114237796278366, 5.27290721488965225350095646853, 5.53923206590883713914504941286, 5.58601007354811792535216873481, 6.29694393117000423096146162170, 6.29861374789186723484136857063, 6.51670678881707777071023036623, 6.69398781065524483932865095223, 7.33216559632879062114481166493

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