Properties

Label 8-588e4-1.1-c5e4-0-9
Degree $8$
Conductor $119538913536$
Sign $1$
Analytic cond. $7.90954\times 10^{7}$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 36·3-s + 810·9-s + 462·11-s + 602·13-s + 228·17-s + 358·19-s + 2.14e3·23-s − 3.52e3·25-s + 1.45e4·27-s − 5.53e3·29-s + 830·31-s + 1.66e4·33-s + 3.91e3·37-s + 2.16e4·39-s + 8.31e3·41-s − 1.45e4·43-s + 4.17e4·47-s + 8.20e3·51-s − 2.21e4·53-s + 1.28e4·57-s + 3.28e4·59-s + 8.37e4·61-s + 8.00e4·67-s + 7.73e4·69-s + 4.47e4·71-s − 2.24e4·73-s − 1.26e5·75-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 1.15·11-s + 0.987·13-s + 0.191·17-s + 0.227·19-s + 0.846·23-s − 1.12·25-s + 3.84·27-s − 1.22·29-s + 0.155·31-s + 2.65·33-s + 0.470·37-s + 2.28·39-s + 0.772·41-s − 1.19·43-s + 2.75·47-s + 0.441·51-s − 1.08·53-s + 0.525·57-s + 1.22·59-s + 2.88·61-s + 2.17·67-s + 1.95·69-s + 1.05·71-s − 0.493·73-s − 2.60·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(7.90954\times 10^{7}\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(52.19958565\)
\(L(\frac12)\) \(\approx\) \(52.19958565\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{4} \)
7 \( 1 \)
good5$C_2 \wr S_4$ \( 1 + 3523 T^{2} - 132 p^{4} T^{3} + 13251176 T^{4} - 132 p^{9} T^{5} + 3523 p^{10} T^{6} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 42 p T + 268705 T^{2} - 50129610 T^{3} + 18647530376 T^{4} - 50129610 p^{5} T^{5} + 268705 p^{10} T^{6} - 42 p^{16} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 602 T - 12679 p T^{2} - 145984794 T^{3} + 357680241104 T^{4} - 145984794 p^{5} T^{5} - 12679 p^{11} T^{6} - 602 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 228 T + 3869284 T^{2} - 1347363564 T^{3} + 6940431119366 T^{4} - 1347363564 p^{5} T^{5} + 3869284 p^{10} T^{6} - 228 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 358 T + 7365277 T^{2} - 2785970626 T^{3} + 25078082046268 T^{4} - 2785970626 p^{5} T^{5} + 7365277 p^{10} T^{6} - 358 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 2148 T + 16852828 T^{2} - 16061095668 T^{3} + 120569983154918 T^{4} - 16061095668 p^{5} T^{5} + 16852828 p^{10} T^{6} - 2148 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5532 T + 56710687 T^{2} + 131401968540 T^{3} + 1151610033241988 T^{4} + 131401968540 p^{5} T^{5} + 56710687 p^{10} T^{6} + 5532 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 830 T + 31255592 T^{2} + 217237417272 T^{3} + 250607576745089 T^{4} + 217237417272 p^{5} T^{5} + 31255592 p^{10} T^{6} - 830 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 3914 T + 81854461 T^{2} - 709188215618 T^{3} + 5749597008366352 T^{4} - 709188215618 p^{5} T^{5} + 81854461 p^{10} T^{6} - 3914 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 8316 T + 131719112 T^{2} + 1793338464780 T^{3} - 11507939827736466 T^{4} + 1793338464780 p^{5} T^{5} + 131719112 p^{10} T^{6} - 8316 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 14518 T + 553805833 T^{2} + 5594375858722 T^{3} + 119245786138973608 T^{4} + 5594375858722 p^{5} T^{5} + 553805833 p^{10} T^{6} + 14518 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 41700 T + 1201404476 T^{2} - 465323341500 p T^{3} + 369215501257269942 T^{4} - 465323341500 p^{6} T^{5} + 1201404476 p^{10} T^{6} - 41700 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 22164 T + 612500711 T^{2} - 9154492772292 T^{3} - 135301388975352444 T^{4} - 9154492772292 p^{5} T^{5} + 612500711 p^{10} T^{6} + 22164 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 32886 T + 2165272693 T^{2} - 53279788989366 T^{3} + 2201676784894672940 T^{4} - 53279788989366 p^{5} T^{5} + 2165272693 p^{10} T^{6} - 32886 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 83732 T + 5138847044 T^{2} - 213873087755964 T^{3} + 7191474746319250502 T^{4} - 213873087755964 p^{5} T^{5} + 5138847044 p^{10} T^{6} - 83732 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 80034 T + 6110296361 T^{2} - 278312842674138 T^{3} + 12284801359836541152 T^{4} - 278312842674138 p^{5} T^{5} + 6110296361 p^{10} T^{6} - 80034 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 44772 T + 5603702956 T^{2} - 219545846683812 T^{3} + 14130153190557486902 T^{4} - 219545846683812 p^{5} T^{5} + 5603702956 p^{10} T^{6} - 44772 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 22470 T + 3732785609 T^{2} + 14345610615750 T^{3} + 8330122733588006676 T^{4} + 14345610615750 p^{5} T^{5} + 3732785609 p^{10} T^{6} + 22470 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 75286 T + 10257886376 T^{2} - 522030297673368 T^{3} + 42333862433009559377 T^{4} - 522030297673368 p^{5} T^{5} + 10257886376 p^{10} T^{6} - 75286 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 17418 T + 11662792993 T^{2} - 252243184391142 T^{3} + 61381737302005449608 T^{4} - 252243184391142 p^{5} T^{5} + 11662792993 p^{10} T^{6} - 17418 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 28944 T + 4985253824 T^{2} - 135699519340656 T^{3} - 6573734910358628322 T^{4} - 135699519340656 p^{5} T^{5} + 4985253824 p^{10} T^{6} - 28944 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 216678 T + 43602999305 T^{2} - 4974916579453302 T^{3} + \)\(56\!\cdots\!00\)\( T^{4} - 4974916579453302 p^{5} T^{5} + 43602999305 p^{10} T^{6} - 216678 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.18102493425257521845906584047, −6.69808121762116658713841796084, −6.36687338852324465801638260491, −6.36598688716242558748899118855, −6.07676048605494948911704554864, −5.65305201763409090172770186779, −5.28771184342674708876054141718, −5.24428039898948773581046069210, −5.03250606626888387137054384718, −4.25695826403004166418053211987, −4.16156170581148714606795259175, −4.07052994801156078545224181445, −4.06828331719991921553191082911, −3.42814849294869845924327700258, −3.28893880922404834078714316081, −3.07503857938357793721555985406, −3.04909450570473859424203583454, −2.11586744594484482600895167391, −2.09448299306356188416019700470, −2.05406851904030609848654173438, −1.89835545859806929760184414904, −1.14794996546507994633999169744, −0.879963474089920637508027276258, −0.64547434166418236436075483797, −0.62230237170248182907945090259, 0.62230237170248182907945090259, 0.64547434166418236436075483797, 0.879963474089920637508027276258, 1.14794996546507994633999169744, 1.89835545859806929760184414904, 2.05406851904030609848654173438, 2.09448299306356188416019700470, 2.11586744594484482600895167391, 3.04909450570473859424203583454, 3.07503857938357793721555985406, 3.28893880922404834078714316081, 3.42814849294869845924327700258, 4.06828331719991921553191082911, 4.07052994801156078545224181445, 4.16156170581148714606795259175, 4.25695826403004166418053211987, 5.03250606626888387137054384718, 5.24428039898948773581046069210, 5.28771184342674708876054141718, 5.65305201763409090172770186779, 6.07676048605494948911704554864, 6.36598688716242558748899118855, 6.36687338852324465801638260491, 6.69808121762116658713841796084, 7.18102493425257521845906584047

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