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Properties

Label 8-1560e4-1.1-c3e4-0-1
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $7.17732\times 10^{7}$
Root an. cond. $9.59390$
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  − 12·3-s − 20·5-s − 11·7-s + 90·9-s + 35·11-s − 52·13-s + 240·15-s − 9·17-s + 32·19-s + 132·21-s − 149·23-s + 250·25-s − 540·27-s + 168·29-s + 280·31-s − 420·33-s + 220·35-s + 37·37-s + 624·39-s + 247·41-s + 132·43-s − 1.80e3·45-s − 517·49-s + 108·51-s + 247·53-s − 700·55-s − 384·57-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s − 0.593·7-s + 10/3·9-s + 0.959·11-s − 1.10·13-s + 4.13·15-s − 0.128·17-s + 0.386·19-s + 1.37·21-s − 1.35·23-s + 2·25-s − 3.84·27-s + 1.07·29-s + 1.62·31-s − 2.21·33-s + 1.06·35-s + 0.164·37-s + 2.56·39-s + 0.940·41-s + 0.468·43-s − 5.96·45-s − 1.50·49-s + 0.296·51-s + 0.640·53-s − 1.71·55-s − 0.892·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(7.17732\times 10^{7}\)
Root analytic conductor: \(9.59390\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 + p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
13$C_1$ \( ( 1 + p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 11 T + 638 T^{2} + 6687 T^{3} + 277250 T^{4} + 6687 p^{3} T^{5} + 638 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 35 T + 3490 T^{2} - 851 p^{2} T^{3} + 6649146 T^{4} - 851 p^{5} T^{5} + 3490 p^{6} T^{6} - 35 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 9 T + 18846 T^{2} + 116583 T^{3} + 136826114 T^{4} + 116583 p^{3} T^{5} + 18846 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 32 T + 10704 T^{2} - 157808 T^{3} + 49128686 T^{4} - 157808 p^{3} T^{5} + 10704 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 149 T + 24036 T^{2} + 2829225 T^{3} + 446878614 T^{4} + 2829225 p^{3} T^{5} + 24036 p^{6} T^{6} + 149 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 168 T + 63100 T^{2} - 13021496 T^{3} + 65556974 p T^{4} - 13021496 p^{3} T^{5} + 63100 p^{6} T^{6} - 168 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 280 T + 102904 T^{2} - 21276568 T^{3} + 4578875278 T^{4} - 21276568 p^{3} T^{5} + 102904 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - p T + 146826 T^{2} - 702159 T^{3} + 9696983914 T^{4} - 702159 p^{3} T^{5} + 146826 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 247 T + 283462 T^{2} - 50131281 T^{3} + 29580506258 T^{4} - 50131281 p^{3} T^{5} + 283462 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 132 T + 249180 T^{2} - 23649572 T^{3} + 27653627094 T^{4} - 23649572 p^{3} T^{5} + 249180 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 171880 T^{2} - 2485664 T^{3} + 14155391086 T^{4} - 2485664 p^{3} T^{5} + 171880 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 247 T + 543922 T^{2} - 97114533 T^{3} + 117321834362 T^{4} - 97114533 p^{3} T^{5} + 543922 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 614 T + 566732 T^{2} - 138319766 T^{3} + 106867828150 T^{4} - 138319766 p^{3} T^{5} + 566732 p^{6} T^{6} - 614 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 719 T + 676314 T^{2} - 315515861 T^{3} + 199831686026 T^{4} - 315515861 p^{3} T^{5} + 676314 p^{6} T^{6} - 719 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 658 T + 433188 T^{2} + 115705454 T^{3} - 53997391258 T^{4} + 115705454 p^{3} T^{5} + 433188 p^{6} T^{6} - 658 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 939 T + 718494 T^{2} - 519988407 T^{3} + 427957789762 T^{4} - 519988407 p^{3} T^{5} + 718494 p^{6} T^{6} - 939 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1452 T + 1742084 T^{2} + 1433964468 T^{3} + 1026387364694 T^{4} + 1433964468 p^{3} T^{5} + 1742084 p^{6} T^{6} + 1452 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 289 T + 1249872 T^{2} - 83356173 T^{3} + 726477052702 T^{4} - 83356173 p^{3} T^{5} + 1249872 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 118 T + 14132 T^{2} - 445321898 T^{3} - 52908707834 T^{4} - 445321898 p^{3} T^{5} + 14132 p^{6} T^{6} + 118 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1319 T + 1221066 T^{2} + 69782097 T^{3} - 879945766 p T^{4} + 69782097 p^{3} T^{5} + 1221066 p^{6} T^{6} + 1319 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 993 T + 2688746 T^{2} + 1621923903 T^{3} + 3070810482858 T^{4} + 1621923903 p^{3} T^{5} + 2688746 p^{6} T^{6} + 993 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.73065889741204450604759331389, −6.54834592837938571333739430747, −6.30247643373145379031681826543, −6.29629820456500572242175762282, −6.05388071744245288702678707126, −5.53395964699676790457322981303, −5.38280416370255515544702380052, −5.31979655437724279948806287512, −5.20816860089348980165750430478, −4.67344099583152164895602241474, −4.59279214085267031994883762332, −4.32519990976052328912840829037, −4.32454820213664077974360030422, −3.95253382604192390448869248035, −3.63515620454189629873188103135, −3.59879317561559448501505582323, −3.45320671983816785270696795292, −2.63636051792285127485543720204, −2.60483631320955072693878786893, −2.46198436323207998472882072740, −2.21322549307073211445083538119, −1.32097446149692544758585933941, −1.15586802919173863936567824391, −1.04808214251369306798675517174, −1.02986780271255580454887609320, 0, 0, 0, 0, 1.02986780271255580454887609320, 1.04808214251369306798675517174, 1.15586802919173863936567824391, 1.32097446149692544758585933941, 2.21322549307073211445083538119, 2.46198436323207998472882072740, 2.60483631320955072693878786893, 2.63636051792285127485543720204, 3.45320671983816785270696795292, 3.59879317561559448501505582323, 3.63515620454189629873188103135, 3.95253382604192390448869248035, 4.32454820213664077974360030422, 4.32519990976052328912840829037, 4.59279214085267031994883762332, 4.67344099583152164895602241474, 5.20816860089348980165750430478, 5.31979655437724279948806287512, 5.38280416370255515544702380052, 5.53395964699676790457322981303, 6.05388071744245288702678707126, 6.29629820456500572242175762282, 6.30247643373145379031681826543, 6.54834592837938571333739430747, 6.73065889741204450604759331389

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