Properties

Label 8-570e4-1.1-c5e4-0-5
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 $4$

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 − 108·7-s − 1.28e3·8-s + 810·9-s − 1.60e3·10-s − 246·11-s − 5.76e3·12-s + 640·13-s + 1.72e3·14-s − 3.60e3·15-s + 8.96e3·16-s − 612·17-s − 1.29e4·18-s + 1.44e3·19-s + 1.60e4·20-s + 3.88e3·21-s + 3.93e3·22-s − 1.24e3·23-s + 4.60e4·24-s + 6.25e3·25-s − 1.02e4·26-s − 1.45e4·27-s − 1.72e4·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 0.833·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s − 0.612·11-s − 11.5·12-s + 1.05·13-s + 2.35·14-s − 4.13·15-s + 35/4·16-s − 0.513·17-s − 9.42·18-s + 0.917·19-s + 8.94·20-s + 1.92·21-s + 1.73·22-s − 0.489·23-s + 16.3·24-s + 2·25-s − 2.97·26-s − 3.84·27-s − 4.16·28-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: \(4\)
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

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_2 \wr S_4$ \( 1 + 108 T + 23900 T^{2} + 1713816 T^{3} + 552560598 T^{4} + 1713816 p^{5} T^{5} + 23900 p^{10} T^{6} + 108 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 246 T + 421460 T^{2} + 80210594 T^{3} + 8369673858 p T^{4} + 80210594 p^{5} T^{5} + 421460 p^{10} T^{6} + 246 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 640 T + 1056956 T^{2} - 361529700 T^{3} + 459852850718 T^{4} - 361529700 p^{5} T^{5} + 1056956 p^{10} T^{6} - 640 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 36 p T + 2493916 T^{2} + 3830078604 T^{3} + 2929546062758 T^{4} + 3830078604 p^{5} T^{5} + 2493916 p^{10} T^{6} + 36 p^{16} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 54 p T + 15057544 T^{2} + 24084920250 T^{3} + 127312782620462 T^{4} + 24084920250 p^{5} T^{5} + 15057544 p^{10} T^{6} + 54 p^{16} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 6230 T + 50227760 T^{2} + 162265143630 T^{3} + 1044362998531878 T^{4} + 162265143630 p^{5} T^{5} + 50227760 p^{10} T^{6} + 6230 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 11360 T + 97774932 T^{2} + 652434470416 T^{3} + 4089333295652182 T^{4} + 652434470416 p^{5} T^{5} + 97774932 p^{10} T^{6} + 11360 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 4792 T + 185999500 T^{2} + 783131830508 T^{3} + 16095708498883886 T^{4} + 783131830508 p^{5} T^{5} + 185999500 p^{10} T^{6} + 4792 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 9170 T + 343922232 T^{2} - 3465660766194 T^{3} + 52702052997241462 T^{4} - 3465660766194 p^{5} T^{5} + 343922232 p^{10} T^{6} - 9170 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 11412 T + 49864308 T^{2} - 2182228289304 T^{3} - 40366792660933082 T^{4} - 2182228289304 p^{5} T^{5} + 49864308 p^{10} T^{6} + 11412 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 29858 T + 908425208 T^{2} + 16046864111154 T^{3} + 297180955468482414 T^{4} + 16046864111154 p^{5} T^{5} + 908425208 p^{10} T^{6} + 29858 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 27498 T + 1596808876 T^{2} - 33463754731734 T^{3} + 988371145337755238 T^{4} - 33463754731734 p^{5} T^{5} + 1596808876 p^{10} T^{6} - 27498 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 54984 T + 3855461692 T^{2} - 124081177325352 T^{3} + 4493870636968603574 T^{4} - 124081177325352 p^{5} T^{5} + 3855461692 p^{10} T^{6} - 54984 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 20868 T + 1289735044 T^{2} - 39241717834460 T^{3} + 1483118740903875366 T^{4} - 39241717834460 p^{5} T^{5} + 1289735044 p^{10} T^{6} - 20868 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 20244 T + 3281754252 T^{2} + 94157149007604 T^{3} + 5199994947231751030 T^{4} + 94157149007604 p^{5} T^{5} + 3281754252 p^{10} T^{6} + 20244 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 86864 T + 9630749420 T^{2} - 493397548429296 T^{3} + 28310999748453748998 T^{4} - 493397548429296 p^{5} T^{5} + 9630749420 p^{10} T^{6} - 86864 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3728 T + 928425484 T^{2} - 26462183539664 T^{3} + 6418662774571597190 T^{4} - 26462183539664 p^{5} T^{5} + 928425484 p^{10} T^{6} + 3728 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 164192 T + 15643001596 T^{2} - 955265739517024 T^{3} + 54103780177126731206 T^{4} - 955265739517024 p^{5} T^{5} + 15643001596 p^{10} T^{6} - 164192 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 60506 T + 8520751864 T^{2} + 352382473085458 T^{3} + 32730072408805508798 T^{4} + 352382473085458 p^{5} T^{5} + 8520751864 p^{10} T^{6} + 60506 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 113798 T + 14967753120 T^{2} - 1344398093521998 T^{3} + \)\(12\!\cdots\!98\)\( T^{4} - 1344398093521998 p^{5} T^{5} + 14967753120 p^{10} T^{6} - 113798 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 79440 T + 33692357804 T^{2} - 1965290512820340 T^{3} + \)\(43\!\cdots\!58\)\( T^{4} - 1965290512820340 p^{5} T^{5} + 33692357804 p^{10} T^{6} - 79440 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.56272577173055801211874233334, −6.88239705305006964599220545433, −6.82492015931688738054624930702, −6.74921631400134490120534337524, −6.73375260143736089650205127208, −6.14790344756453748547629432099, −6.03754677328473983322164165738, −5.95007657043275928242283583279, −5.80430050183295128265861166432, −5.17982055632921530602193884602, −5.14266377876641583340562290282, −5.03267713282425962922143212345, −4.97874486878879759053027827247, −3.87688193900449413936159254413, −3.69911210144269574218698671211, −3.61009421495241861517796178312, −3.52125773833281939904756082588, −2.53094550363241591353237435252, −2.45802004752113121626778136770, −2.27376408918062841460362911790, −2.03478606288426683579999800418, −1.43225706458031069663282292998, −1.23796139380805021041356245530, −1.08676122400240349633269842782, −1.04917506594246080307402499108, 0, 0, 0, 0, 1.04917506594246080307402499108, 1.08676122400240349633269842782, 1.23796139380805021041356245530, 1.43225706458031069663282292998, 2.03478606288426683579999800418, 2.27376408918062841460362911790, 2.45802004752113121626778136770, 2.53094550363241591353237435252, 3.52125773833281939904756082588, 3.61009421495241861517796178312, 3.69911210144269574218698671211, 3.87688193900449413936159254413, 4.97874486878879759053027827247, 5.03267713282425962922143212345, 5.14266377876641583340562290282, 5.17982055632921530602193884602, 5.80430050183295128265861166432, 5.95007657043275928242283583279, 6.03754677328473983322164165738, 6.14790344756453748547629432099, 6.73375260143736089650205127208, 6.74921631400134490120534337524, 6.82492015931688738054624930702, 6.88239705305006964599220545433, 7.56272577173055801211874233334

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