Properties

Label 8-588e4-1.1-c3e4-0-3
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 $4$

Origins

Origins of factors

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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: induced by $\chi_{588} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
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

−7.86315339390961880954548378579, −7.20350869267404163775536367343, −7.09350741436190570309337792608, −6.85173273663045907348056130972, −6.85099992104691433400277791690, −6.58998219620040408884994300913, −6.25312991665996061356625204666, −6.07832588411505786965066831512, −6.02142619955825502294064851900, −5.41916105093184054402285194947, −5.37026543394511362096076561078, −5.12222952160649606539602335545, −4.97825154854960192309880990932, −4.38687920095016036528331189445, −4.35988837380580326433221385717, −4.22873820053508337683882718918, −4.11723246662222883427579403099, −3.33342757741722071905706309588, −3.11725689738771231029012043206, −2.77817681564899282027104408059, −2.66521454988000789994205161543, −1.72924220673063275512560032672, −1.63549290811764027845480793228, −1.39361223516768465927923379413, −1.27372297138780187910107890154, 0, 0, 0, 0, 1.27372297138780187910107890154, 1.39361223516768465927923379413, 1.63549290811764027845480793228, 1.72924220673063275512560032672, 2.66521454988000789994205161543, 2.77817681564899282027104408059, 3.11725689738771231029012043206, 3.33342757741722071905706309588, 4.11723246662222883427579403099, 4.22873820053508337683882718918, 4.35988837380580326433221385717, 4.38687920095016036528331189445, 4.97825154854960192309880990932, 5.12222952160649606539602335545, 5.37026543394511362096076561078, 5.41916105093184054402285194947, 6.02142619955825502294064851900, 6.07832588411505786965066831512, 6.25312991665996061356625204666, 6.58998219620040408884994300913, 6.85099992104691433400277791690, 6.85173273663045907348056130972, 7.09350741436190570309337792608, 7.20350869267404163775536367343, 7.86315339390961880954548378579

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