Properties

Label 8-570e4-1.1-c5e4-0-0
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $6.98460\times 10^{7}$
Root an. cond. $9.56131$
Motivic weight $5$
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  − 16·2-s − 36·3-s + 160·4-s − 100·5-s + 576·6-s − 88·7-s − 1.28e3·8-s + 810·9-s + 1.60e3·10-s + 940·11-s − 5.76e3·12-s − 34·13-s + 1.40e3·14-s + 3.60e3·15-s + 8.96e3·16-s − 138·17-s − 1.29e4·18-s + 1.44e3·19-s − 1.60e4·20-s + 3.16e3·21-s − 1.50e4·22-s + 486·23-s + 4.60e4·24-s + 6.25e3·25-s + 544·26-s − 1.45e4·27-s − 1.40e4·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s − 0.678·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s + 2.34·11-s − 11.5·12-s − 0.0557·13-s + 1.91·14-s + 4.13·15-s + 35/4·16-s − 0.115·17-s − 9.42·18-s + 0.917·19-s − 8.94·20-s + 1.56·21-s − 6.62·22-s + 0.191·23-s + 16.3·24-s + 2·25-s + 0.157·26-s − 3.84·27-s − 3.39·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(6.98460\times 10^{7}\)
Root analytic conductor: \(9.56131\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.3258896393\)
\(L(\frac12)\) \(\approx\) \(0.3258896393\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
3$C_1$ \( ( 1 + p^{2} T )^{4} \)
5$C_1$ \( ( 1 + p^{2} T )^{4} \)
19$C_1$ \( ( 1 - p^{2} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 88 T + 37390 T^{2} + 5193896 T^{3} + 703224938 T^{4} + 5193896 p^{5} T^{5} + 37390 p^{10} T^{6} + 88 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 940 T + 69106 p T^{2} - 397585932 T^{3} + 188976480690 T^{4} - 397585932 p^{5} T^{5} + 69106 p^{11} T^{6} - 940 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 34 T + 299446 T^{2} + 29795562 T^{3} + 284149196298 T^{4} + 29795562 p^{5} T^{5} + 299446 p^{10} T^{6} + 34 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 138 T + 1594220 T^{2} + 1249745006 T^{3} + 3312312999318 T^{4} + 1249745006 p^{5} T^{5} + 1594220 p^{10} T^{6} + 138 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 486 T + 6993464 T^{2} - 4526395318 T^{3} + 92179087459278 T^{4} - 4526395318 p^{5} T^{5} + 6993464 p^{10} T^{6} - 486 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 3296 T + 38800710 T^{2} - 169169018976 T^{3} + 939071456387698 T^{4} - 169169018976 p^{5} T^{5} + 38800710 p^{10} T^{6} - 3296 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 6686 T + 82518912 T^{2} + 458403843262 T^{3} + 3322381328138878 T^{4} + 458403843262 p^{5} T^{5} + 82518912 p^{10} T^{6} + 6686 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 14218 T + 286189078 T^{2} + 2527537097474 T^{3} + 29297216864348714 T^{4} + 2527537097474 p^{5} T^{5} + 286189078 p^{10} T^{6} + 14218 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 1342 T + 315117818 T^{2} - 449470825854 T^{3} + 46235700404373714 T^{4} - 449470825854 p^{5} T^{5} + 315117818 p^{10} T^{6} - 1342 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 26330 T + 790381722 T^{2} + 11793104986150 T^{3} + 187041147469891738 T^{4} + 11793104986150 p^{5} T^{5} + 790381722 p^{10} T^{6} + 26330 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 7010 T + 673144040 T^{2} + 4001045889282 T^{3} + 208598439638225742 T^{4} + 4001045889282 p^{5} T^{5} + 673144040 p^{10} T^{6} + 7010 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 22782 T + 1687454996 T^{2} + 27144665483506 T^{3} + 1058799638354350038 T^{4} + 27144665483506 p^{5} T^{5} + 1687454996 p^{10} T^{6} + 22782 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 13358 T + 831362588 T^{2} + 2500415162526 T^{3} + 798638634074957622 T^{4} + 2500415162526 p^{5} T^{5} + 831362588 p^{10} T^{6} + 13358 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 16008 T + 3459805664 T^{2} - 40702186750680 T^{3} + 4417670139231900846 T^{4} - 40702186750680 p^{5} T^{5} + 3459805664 p^{10} T^{6} - 16008 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 22696 T + 1028454940 T^{2} - 9930394127256 T^{3} - 1439898412074337578 T^{4} - 9930394127256 p^{5} T^{5} + 1028454940 p^{10} T^{6} + 22696 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 52584 T + 7980195140 T^{2} - 286825047239816 T^{3} + 22287425986400570838 T^{4} - 286825047239816 p^{5} T^{5} + 7980195140 p^{10} T^{6} - 52584 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 101048 T + 8221334284 T^{2} + 489343012130056 T^{3} + 23661531366700914950 T^{4} + 489343012130056 p^{5} T^{5} + 8221334284 p^{10} T^{6} + 101048 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 113090 T + 16005469996 T^{2} + 1084923235248730 T^{3} + 79706142439766914406 T^{4} + 1084923235248730 p^{5} T^{5} + 16005469996 p^{10} T^{6} + 113090 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 65384 T + 6718730500 T^{2} + 35229356986312 T^{3} + 8480019068136866198 T^{4} + 35229356986312 p^{5} T^{5} + 6718730500 p^{10} T^{6} + 65384 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 97354 T + 13216650482 T^{2} - 1214576660224362 T^{3} + \)\(10\!\cdots\!14\)\( T^{4} - 1214576660224362 p^{5} T^{5} + 13216650482 p^{10} T^{6} - 97354 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 198934 T + 35423084782 T^{2} + 4247926764541894 T^{3} + \)\(46\!\cdots\!50\)\( T^{4} + 4247926764541894 p^{5} T^{5} + 35423084782 p^{10} T^{6} + 198934 p^{15} T^{7} + p^{20} T^{8} \)
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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

−6.93602464989096327073657692022, −6.67884645359082151334462989013, −6.58156873357932077037621243691, −6.46691485421904193887525725024, −6.43981667502678933364910796415, −5.76302336806073178306341762779, −5.47569705585849087520286837734, −5.47200814550344151890154506355, −5.25590790263357167506561845666, −4.54892940374298598621130430294, −4.37262144756782869309085150541, −4.32093020165520252133972206173, −3.95771650768883332522073245765, −3.40291540628465690194466327935, −3.20817195695231306757808044077, −3.08317403865466998866041715502, −2.98897877158509948128742994174, −1.80562036304419735133363815652, −1.74707286055795662914597954084, −1.64980369824410100090104980436, −1.42402137684595718690465214794, −0.885527989164755017083998010960, −0.43808037305316715281099766231, −0.43353352370111677893852619581, −0.38869611586015711434720167622, 0.38869611586015711434720167622, 0.43353352370111677893852619581, 0.43808037305316715281099766231, 0.885527989164755017083998010960, 1.42402137684595718690465214794, 1.64980369824410100090104980436, 1.74707286055795662914597954084, 1.80562036304419735133363815652, 2.98897877158509948128742994174, 3.08317403865466998866041715502, 3.20817195695231306757808044077, 3.40291540628465690194466327935, 3.95771650768883332522073245765, 4.32093020165520252133972206173, 4.37262144756782869309085150541, 4.54892940374298598621130430294, 5.25590790263357167506561845666, 5.47200814550344151890154506355, 5.47569705585849087520286837734, 5.76302336806073178306341762779, 6.43981667502678933364910796415, 6.46691485421904193887525725024, 6.58156873357932077037621243691, 6.67884645359082151334462989013, 6.93602464989096327073657692022

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