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Properties

Label 8-684e4-1.1-c5e4-0-3
Degree $8$
Conductor $218889236736$
Sign $1$
Analytic cond. $1.44832\times 10^{8}$
Root an. cond. $10.4738$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 110·5-s + 30·7-s − 706·11-s + 788·13-s − 240·17-s − 1.44e3·19-s − 5.88e3·23-s + 5.68e3·25-s − 5.24e3·29-s − 860·31-s + 3.30e3·35-s − 2.07e4·37-s + 1.02e4·41-s − 1.25e4·43-s + 4.82e3·47-s − 4.38e4·49-s + 7.64e4·53-s − 7.76e4·55-s − 2.38e4·59-s − 3.24e4·61-s + 8.66e4·65-s + 5.02e3·67-s − 1.21e5·71-s − 1.04e5·73-s − 2.11e4·77-s + 1.17e5·79-s − 9.28e4·83-s + ⋯
L(s)  = 1  + 1.96·5-s + 0.231·7-s − 1.75·11-s + 1.29·13-s − 0.201·17-s − 0.917·19-s − 2.31·23-s + 1.81·25-s − 1.15·29-s − 0.160·31-s + 0.455·35-s − 2.48·37-s + 0.948·41-s − 1.03·43-s + 0.318·47-s − 2.60·49-s + 3.74·53-s − 3.46·55-s − 0.893·59-s − 1.11·61-s + 2.54·65-s + 0.136·67-s − 2.85·71-s − 2.29·73-s − 0.407·77-s + 2.11·79-s − 1.47·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.44832\times 10^{8}\)
Root analytic conductor: \(10.4738\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
3 \( 1 \)
19$C_1$ \( ( 1 + p^{2} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 22 p T + 6413 T^{2} - 8878 p^{2} T^{3} + 5767376 T^{4} - 8878 p^{7} T^{5} + 6413 p^{10} T^{6} - 22 p^{16} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 30 T + 44752 T^{2} + 98856 p T^{3} + 881393049 T^{4} + 98856 p^{6} T^{5} + 44752 p^{10} T^{6} - 30 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 706 T + 658673 T^{2} + 323249858 T^{3} + 161127831824 T^{4} + 323249858 p^{5} T^{5} + 658673 p^{10} T^{6} + 706 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 788 T + 1186231 T^{2} - 700956980 T^{3} + 620682637204 T^{4} - 700956980 p^{5} T^{5} + 1186231 p^{10} T^{6} - 788 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 240 T + 1613282 T^{2} + 761498208 T^{3} + 1482281900259 T^{4} + 761498208 p^{5} T^{5} + 1613282 p^{10} T^{6} + 240 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 5884 T + 21411695 T^{2} + 54686158844 T^{3} + 123442690709456 T^{4} + 54686158844 p^{5} T^{5} + 21411695 p^{10} T^{6} + 5884 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5240 T + 60856211 T^{2} + 253837964428 T^{3} + 1617188633134184 T^{4} + 253837964428 p^{5} T^{5} + 60856211 p^{10} T^{6} + 5240 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 860 T + 91400044 T^{2} + 62709027884 T^{3} + 3723403118650534 T^{4} + 62709027884 p^{5} T^{5} + 91400044 p^{10} T^{6} + 860 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 20732 T + 227807476 T^{2} + 1980389778452 T^{3} + 17275967393509126 T^{4} + 1980389778452 p^{5} T^{5} + 227807476 p^{10} T^{6} + 20732 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 10204 T + 388738856 T^{2} - 3038996205812 T^{3} + 64399227505813550 T^{4} - 3038996205812 p^{5} T^{5} + 388738856 p^{10} T^{6} - 10204 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 12554 T + 479648677 T^{2} + 4138530607910 T^{3} + 95434697410657180 T^{4} + 4138530607910 p^{5} T^{5} + 479648677 p^{10} T^{6} + 12554 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 4826 T + 630393053 T^{2} - 4163406192922 T^{3} + 185783159690442980 T^{4} - 4163406192922 p^{5} T^{5} + 630393053 p^{10} T^{6} - 4826 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 76484 T + 3631277615 T^{2} - 116460309293524 T^{3} + 2762604272081552756 T^{4} - 116460309293524 p^{5} T^{5} + 3631277615 p^{10} T^{6} - 76484 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 23898 T + 1720660361 T^{2} + 24733920983166 T^{3} + 1308640952006846328 T^{4} + 24733920983166 p^{5} T^{5} + 1720660361 p^{10} T^{6} + 23898 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 32482 T + 2031335641 T^{2} + 31965315437422 T^{3} + 1619420039494417348 T^{4} + 31965315437422 p^{5} T^{5} + 2031335641 p^{10} T^{6} + 32482 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5022 T + 984310405 T^{2} - 20898766206198 T^{3} - 478919880995889060 T^{4} - 20898766206198 p^{5} T^{5} + 984310405 p^{10} T^{6} - 5022 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 121300 T + 12237102908 T^{2} + 737604792352148 T^{3} + 38052926318924245718 T^{4} + 737604792352148 p^{5} T^{5} + 12237102908 p^{10} T^{6} + 121300 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 104700 T + 8117942698 T^{2} + 501529670831376 T^{3} + 26088351394045555539 T^{4} + 501529670831376 p^{5} T^{5} + 8117942698 p^{10} T^{6} + 104700 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 117128 T + 14278334536 T^{2} - 951853078254248 T^{3} + 65526252116476751662 T^{4} - 951853078254248 p^{5} T^{5} + 14278334536 p^{10} T^{6} - 117128 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 92832 T + 13268062568 T^{2} + 894150738137088 T^{3} + 71415811204234063134 T^{4} + 894150738137088 p^{5} T^{5} + 13268062568 p^{10} T^{6} + 92832 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 5988 T - 2589975052 T^{2} - 17516631078900 T^{3} + 49528889947005313350 T^{4} - 17516631078900 p^{5} T^{5} - 2589975052 p^{10} T^{6} + 5988 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 22972 T + 12158221672 T^{2} - 310772507916916 T^{3} + \)\(18\!\cdots\!82\)\( T^{4} - 310772507916916 p^{5} T^{5} + 12158221672 p^{10} T^{6} - 22972 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.32547427696961098964618322242, −6.81182613526223781133570872902, −6.56903321195575120154886164823, −6.44767708769546957141803136207, −6.33161506994437686320396618427, −5.95398970467190190342441440207, −5.73727349700779493519504251164, −5.60754077043738996732063301395, −5.45385117200077294756647350060, −5.08078255518361660365427284538, −5.04730230870815515425792587412, −4.55737205848061629755373526427, −4.41165315433804647457512403143, −3.89066179616369162831925739901, −3.81997024705017380634096751438, −3.54546239632540110426404819866, −3.33212033486915248513913483263, −2.63211849248349838203425924493, −2.53941169863770374595426984863, −2.33946653448576190171848282646, −2.31574844510367936883585455729, −1.54083994032539700368871739180, −1.48017584249023091116751964373, −1.47415808387866724314393913405, −1.13584083475813878773685534143, 0, 0, 0, 0, 1.13584083475813878773685534143, 1.47415808387866724314393913405, 1.48017584249023091116751964373, 1.54083994032539700368871739180, 2.31574844510367936883585455729, 2.33946653448576190171848282646, 2.53941169863770374595426984863, 2.63211849248349838203425924493, 3.33212033486915248513913483263, 3.54546239632540110426404819866, 3.81997024705017380634096751438, 3.89066179616369162831925739901, 4.41165315433804647457512403143, 4.55737205848061629755373526427, 5.04730230870815515425792587412, 5.08078255518361660365427284538, 5.45385117200077294756647350060, 5.60754077043738996732063301395, 5.73727349700779493519504251164, 5.95398970467190190342441440207, 6.33161506994437686320396618427, 6.44767708769546957141803136207, 6.56903321195575120154886164823, 6.81182613526223781133570872902, 7.32547427696961098964618322242

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