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Properties

Label 8-1014e4-1.1-c3e4-0-4
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $1.28119\times 10^{7}$
Root an. cond. $7.73485$
Motivic weight $3$
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  − 8·2-s − 12·3-s + 40·4-s − 8·5-s + 96·6-s − 22·7-s − 160·8-s + 90·9-s + 64·10-s + 18·11-s − 480·12-s + 176·14-s + 96·15-s + 560·16-s − 76·17-s − 720·18-s − 102·19-s − 320·20-s + 264·21-s − 144·22-s + 182·23-s + 1.92e3·24-s − 77·25-s − 540·27-s − 880·28-s + 38·29-s − 768·30-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s − 0.715·5-s + 6.53·6-s − 1.18·7-s − 7.07·8-s + 10/3·9-s + 2.02·10-s + 0.493·11-s − 11.5·12-s + 3.35·14-s + 1.65·15-s + 35/4·16-s − 1.08·17-s − 9.42·18-s − 1.23·19-s − 3.57·20-s + 2.74·21-s − 1.39·22-s + 1.64·23-s + 16.3·24-s − 0.615·25-s − 3.84·27-s − 5.93·28-s + 0.243·29-s − 4.67·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.28119\times 10^{7}\)
Root analytic conductor: \(7.73485\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
3$C_1$ \( ( 1 + p T )^{4} \)
13 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 141 T^{2} - 144 p T^{3} + 5332 T^{4} - 144 p^{4} T^{5} + 141 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 887 T^{2} + 11820 T^{3} + 336950 T^{4} + 11820 p^{3} T^{5} + 887 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 3422 T^{2} - 37098 T^{3} + 5604474 T^{4} - 37098 p^{3} T^{5} + 3422 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 76 T + 13101 T^{2} + 802788 T^{3} + 95229256 T^{4} + 802788 p^{3} T^{5} + 13101 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 102 T + 24118 T^{2} + 1682046 T^{3} + 229660266 T^{4} + 1682046 p^{3} T^{5} + 24118 p^{6} T^{6} + 102 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 182 T + 40902 T^{2} - 4539630 T^{3} + 629073994 T^{4} - 4539630 p^{3} T^{5} + 40902 p^{6} T^{6} - 182 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 38 T + 37929 T^{2} + 5042562 T^{3} + 451344412 T^{4} + 5042562 p^{3} T^{5} + 37929 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 188 T + 32081 T^{2} - 614808 T^{3} - 248987596 T^{4} - 614808 p^{3} T^{5} + 32081 p^{6} T^{6} + 188 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 506 T + 246689 T^{2} + 65883078 T^{3} + 18354809000 T^{4} + 65883078 p^{3} T^{5} + 246689 p^{6} T^{6} + 506 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 158 T + 114165 T^{2} - 13187802 T^{3} + 7268320456 T^{4} - 13187802 p^{3} T^{5} + 114165 p^{6} T^{6} - 158 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 456 T + 219575 T^{2} + 63531090 T^{3} + 19177709838 T^{4} + 63531090 p^{3} T^{5} + 219575 p^{6} T^{6} + 456 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 942 T + 683078 T^{2} - 318147246 T^{3} + 120491521386 T^{4} - 318147246 p^{3} T^{5} + 683078 p^{6} T^{6} - 942 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 606 T + 541901 T^{2} - 200719530 T^{3} + 110880402960 T^{4} - 200719530 p^{3} T^{5} + 541901 p^{6} T^{6} - 606 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 392 T + 477168 T^{2} + 33994440 T^{3} + 82865273998 T^{4} + 33994440 p^{3} T^{5} + 477168 p^{6} T^{6} + 392 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 6 p T + 505274 T^{2} + 229803816 T^{3} + 144630853863 T^{4} + 229803816 p^{3} T^{5} + 505274 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 434 T + 611495 T^{2} + 284894892 T^{3} + 208774472366 T^{4} + 284894892 p^{3} T^{5} + 611495 p^{6} T^{6} + 434 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1110 T + 767198 T^{2} + 207267606 T^{3} + 98071512762 T^{4} + 207267606 p^{3} T^{5} + 767198 p^{6} T^{6} + 1110 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 952 T + 1040102 T^{2} - 811465920 T^{3} + 610533649019 T^{4} - 811465920 p^{3} T^{5} + 1040102 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 962 T + 981365 T^{2} - 836088342 T^{3} + 850151443532 T^{4} - 836088342 p^{3} T^{5} + 981365 p^{6} T^{6} - 962 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 502 T + 1063794 T^{2} - 702023814 T^{3} + 574083090874 T^{4} - 702023814 p^{3} T^{5} + 1063794 p^{6} T^{6} - 502 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1980 T + 4027772 T^{2} + 4413486996 T^{3} + 4697978951190 T^{4} + 4413486996 p^{3} T^{5} + 4027772 p^{6} T^{6} + 1980 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 972 T + 3929869 T^{2} + 2688729120 T^{3} + 5508134347788 T^{4} + 2688729120 p^{3} T^{5} + 3929869 p^{6} T^{6} + 972 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.26893195889878047545862289179, −6.86737014576500088170611767197, −6.77434948449968113999360817329, −6.74337214542783409051717230751, −6.45844023425768133162565811760, −6.03195826319598335534417860295, −5.98609050815063704837109884887, −5.84219388871508770819853044269, −5.79043232506690207420931693566, −5.06254373428794541867153252552, −4.87692784332118458652827414079, −4.84209415589047453578646161816, −4.66398469535290230372782416354, −3.90317345533111410824075794972, −3.76363017439090009955949839682, −3.70305685889634720934088194245, −3.49210276374541321362804059036, −2.85005459444147881535998144545, −2.66961784831253597025777870513, −2.22236589020936104586212796909, −2.09408076688975720274091653408, −1.58183771256513781527025315794, −1.36695698647025976350854331807, −0.939223346154471092078284638605, −0.877061125563815650514238360822, 0, 0, 0, 0, 0.877061125563815650514238360822, 0.939223346154471092078284638605, 1.36695698647025976350854331807, 1.58183771256513781527025315794, 2.09408076688975720274091653408, 2.22236589020936104586212796909, 2.66961784831253597025777870513, 2.85005459444147881535998144545, 3.49210276374541321362804059036, 3.70305685889634720934088194245, 3.76363017439090009955949839682, 3.90317345533111410824075794972, 4.66398469535290230372782416354, 4.84209415589047453578646161816, 4.87692784332118458652827414079, 5.06254373428794541867153252552, 5.79043232506690207420931693566, 5.84219388871508770819853044269, 5.98609050815063704837109884887, 6.03195826319598335534417860295, 6.45844023425768133162565811760, 6.74337214542783409051717230751, 6.77434948449968113999360817329, 6.86737014576500088170611767197, 7.26893195889878047545862289179

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