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Properties

Label 8-588e4-1.1-c3e4-0-1
Degree $8$
Conductor $119538913536$
Sign $1$
Analytic cond. $1.44868\times 10^{6}$
Root an. cond. $5.89008$
Motivic weight $3$
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  + 12·3-s + 90·9-s + 48·17-s + 192·19-s + 192·23-s − 88·25-s + 540·27-s + 96·29-s + 48·31-s + 256·37-s + 1.00e3·41-s − 112·43-s + 864·47-s + 576·51-s − 648·53-s + 2.30e3·57-s + 336·59-s + 960·61-s + 720·67-s + 2.30e3·69-s − 1.34e3·71-s + 672·73-s − 1.05e3·75-s − 1.98e3·79-s + 2.83e3·81-s + 3.12e3·83-s + 1.15e3·87-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 0.684·17-s + 2.31·19-s + 1.74·23-s − 0.703·25-s + 3.84·27-s + 0.614·29-s + 0.278·31-s + 1.13·37-s + 3.83·41-s − 0.397·43-s + 2.68·47-s + 1.58·51-s − 1.67·53-s + 5.35·57-s + 0.741·59-s + 2.01·61-s + 1.31·67-s + 4.01·69-s − 2.24·71-s + 1.07·73-s − 1.62·75-s − 2.82·79-s + 35/9·81-s + 4.12·83-s + 1.41·87-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.44868\times 10^{6}\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(33.87694468\)
\(L(\frac12)\) \(\approx\) \(33.87694468\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 88 T^{2} - 576 T^{3} + 18498 T^{4} - 576 p^{3} T^{5} + 88 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 108 p T^{2} + 82944 T^{3} + 352406 T^{4} + 82944 p^{3} T^{5} + 108 p^{7} T^{6} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 1696 T^{2} - 31104 T^{3} + 7806642 T^{4} - 31104 p^{3} T^{5} + 1696 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 9880 T^{2} - 116976 T^{3} + 43940658 T^{4} - 116976 p^{3} T^{5} + 9880 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 24156 T^{2} - 2400960 T^{3} + 195309110 T^{4} - 2400960 p^{3} T^{5} + 24156 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 52596 T^{2} - 6099648 T^{3} + 943244678 T^{4} - 6099648 p^{3} T^{5} + 52596 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 96 T + 85860 T^{2} - 6375072 T^{3} + 3037151990 T^{4} - 6375072 p^{3} T^{5} + 85860 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 75532 T^{2} + 1415952 T^{3} + 2535445158 T^{4} + 1415952 p^{3} T^{5} + 75532 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 256 T + 66772 T^{2} + 2259200 T^{3} + 153679222 T^{4} + 2259200 p^{3} T^{5} + 66772 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 1008 T + 609784 T^{2} - 250033968 T^{3} + 76165467474 T^{4} - 250033968 p^{3} T^{5} + 609784 p^{6} T^{6} - 1008 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 112 T + 246988 T^{2} + 11722480 T^{3} + 25842449782 T^{4} + 11722480 p^{3} T^{5} + 246988 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 864 T + 642220 T^{2} - 282465504 T^{3} + 110627505318 T^{4} - 282465504 p^{3} T^{5} + 642220 p^{6} T^{6} - 864 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 648 T + 386892 T^{2} + 114656472 T^{3} + 50365144694 T^{4} + 114656472 p^{3} T^{5} + 386892 p^{6} T^{6} + 648 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 336 T + 534364 T^{2} - 159356496 T^{3} + 154119406422 T^{4} - 159356496 p^{3} T^{5} + 534364 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 960 T + 577728 T^{2} - 153988800 T^{3} + 52569538418 T^{4} - 153988800 p^{3} T^{5} + 577728 p^{6} T^{6} - 960 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 720 T + 533740 T^{2} - 318964176 T^{3} + 154478344470 T^{4} - 318964176 p^{3} T^{5} + 533740 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1344 T + 1892084 T^{2} + 1427151168 T^{3} + 1080205217862 T^{4} + 1427151168 p^{3} T^{5} + 1892084 p^{6} T^{6} + 1344 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 672 T + 1088640 T^{2} - 325175712 T^{3} + 453343664738 T^{4} - 325175712 p^{3} T^{5} + 1088640 p^{6} T^{6} - 672 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1984 T + 1725436 T^{2} + 666213568 T^{3} + 202189693510 T^{4} + 666213568 p^{3} T^{5} + 1725436 p^{6} T^{6} + 1984 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3120 T + 5531020 T^{2} - 6555475248 T^{3} + 5745505983510 T^{4} - 6555475248 p^{3} T^{5} + 5531020 p^{6} T^{6} - 3120 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 2160 T + 3343000 T^{2} - 3573514800 T^{3} + 3407992760850 T^{4} - 3573514800 p^{3} T^{5} + 3343000 p^{6} T^{6} - 2160 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 2016 T + 2898432 T^{2} - 2840274144 T^{3} + 3044116636418 T^{4} - 2840274144 p^{3} T^{5} + 2898432 p^{6} T^{6} - 2016 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.52685919045343486110822037873, −7.26364555242624219554171899662, −6.83956464623397615038187838264, −6.68251914610769123411660102763, −6.56169152503891027324090656084, −5.91876082643510898155775867560, −5.76659530442545319619885413370, −5.71590782311141448480120028254, −5.38413425351335828253403332730, −4.81352506681828629651156522151, −4.70935503286620181948025040039, −4.52400952071025887119104238588, −4.30848289549530939696449049009, −3.70771820213003901402818259259, −3.54636621061499575506738359687, −3.34685138088927157094285242409, −3.32462742399688347107116939878, −2.71743087310872876435127422539, −2.44211069902644343089517227232, −2.26601637140750958435337990185, −2.24050293296710511279718460527, −1.26774062652373213841259051934, −1.14353547864860551483653245678, −0.855674291971716201627547145745, −0.66849252999246936064228612097, 0.66849252999246936064228612097, 0.855674291971716201627547145745, 1.14353547864860551483653245678, 1.26774062652373213841259051934, 2.24050293296710511279718460527, 2.26601637140750958435337990185, 2.44211069902644343089517227232, 2.71743087310872876435127422539, 3.32462742399688347107116939878, 3.34685138088927157094285242409, 3.54636621061499575506738359687, 3.70771820213003901402818259259, 4.30848289549530939696449049009, 4.52400952071025887119104238588, 4.70935503286620181948025040039, 4.81352506681828629651156522151, 5.38413425351335828253403332730, 5.71590782311141448480120028254, 5.76659530442545319619885413370, 5.91876082643510898155775867560, 6.56169152503891027324090656084, 6.68251914610769123411660102763, 6.83956464623397615038187838264, 7.26364555242624219554171899662, 7.52685919045343486110822037873

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