Properties

Label 8-570e4-1.1-c5e4-0-2
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 36·3-s + 160·4-s − 100·5-s − 576·6-s + 26·7-s − 1.28e3·8-s + 810·9-s + 1.60e3·10-s − 472·11-s + 5.76e3·12-s − 482·13-s − 416·14-s − 3.60e3·15-s + 8.96e3·16-s + 1.81e3·17-s − 1.29e4·18-s − 1.44e3·19-s − 1.60e4·20-s + 936·21-s + 7.55e3·22-s − 418·23-s − 4.60e4·24-s + 6.25e3·25-s + 7.71e3·26-s + 1.45e4·27-s + 4.16e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s + 0.200·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s − 1.17·11-s + 11.5·12-s − 0.791·13-s − 0.567·14-s − 4.13·15-s + 35/4·16-s + 1.52·17-s − 9.42·18-s − 0.917·19-s − 8.94·20-s + 0.463·21-s + 3.32·22-s − 0.164·23-s − 16.3·24-s + 2·25-s + 2.23·26-s + 3.84·27-s + 1.00·28-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

\(L(3)\) \(\approx\) \(3.792609452\)
\(L(\frac12)\) \(\approx\) \(3.792609452\)
\(L(\frac{7}{2})\) 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$C_2 \wr S_4$ \( 1 - 26 T + 32878 T^{2} + 329234 T^{3} + 634435394 T^{4} + 329234 p^{5} T^{5} + 32878 p^{10} T^{6} - 26 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 472 T + 528642 T^{2} + 214392840 T^{3} + 120946932922 T^{4} + 214392840 p^{5} T^{5} + 528642 p^{10} T^{6} + 472 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 482 T - 194910 T^{2} - 108002150 T^{3} + 93279817234 T^{4} - 108002150 p^{5} T^{5} - 194910 p^{10} T^{6} + 482 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 1816 T + 4112840 T^{2} - 4319213000 T^{3} + 6410095010414 T^{4} - 4319213000 p^{5} T^{5} + 4112840 p^{10} T^{6} - 1816 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 418 T + 23655808 T^{2} + 9310107122 T^{3} + 221449581032990 T^{4} + 9310107122 p^{5} T^{5} + 23655808 p^{10} T^{6} + 418 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 10396 T + 96816618 T^{2} + 523602302340 T^{3} + 2856223935245482 T^{4} + 523602302340 p^{5} T^{5} + 96816618 p^{10} T^{6} + 10396 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 528 T + 22600840 T^{2} + 172936791776 T^{3} - 475480400959122 T^{4} + 172936791776 p^{5} T^{5} + 22600840 p^{10} T^{6} - 528 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 5774 T + 133142626 T^{2} + 284483166586 T^{3} + 5083380624173266 T^{4} + 284483166586 p^{5} T^{5} + 133142626 p^{10} T^{6} - 5774 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 9620 T + 389782970 T^{2} - 2354535409340 T^{3} + 61079352777640618 T^{4} - 2354535409340 p^{5} T^{5} + 389782970 p^{10} T^{6} - 9620 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 21098 T + 63822022 T^{2} - 1579491260934 T^{3} + 52142824197803778 T^{4} - 1579491260934 p^{5} T^{5} + 63822022 p^{10} T^{6} - 21098 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 18858 T + 447301904 T^{2} + 3870285508458 T^{3} + 79501635369371358 T^{4} + 3870285508458 p^{5} T^{5} + 447301904 p^{10} T^{6} + 18858 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 822 T + 764763668 T^{2} - 272798189658 T^{3} + 476635006838781270 T^{4} - 272798189658 p^{5} T^{5} + 764763668 p^{10} T^{6} - 822 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 28672 T + 1586169972 T^{2} - 9813018541056 T^{3} + 809050710978133894 T^{4} - 9813018541056 p^{5} T^{5} + 1586169972 p^{10} T^{6} - 28672 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 77748 T + 5039475784 T^{2} - 203905717895276 T^{3} + 7089319458482892030 T^{4} - 203905717895276 p^{5} T^{5} + 5039475784 p^{10} T^{6} - 77748 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 82400 T + 6879018684 T^{2} - 312402056706784 T^{3} + 14401137257434002070 T^{4} - 312402056706784 p^{5} T^{5} + 6879018684 p^{10} T^{6} - 82400 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 928 T + 4010986740 T^{2} - 58042258986912 T^{3} + 7965000374974375798 T^{4} - 58042258986912 p^{5} T^{5} + 4010986740 p^{10} T^{6} - 928 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 121100 T + 13631111316 T^{2} - 853752814863508 T^{3} + 48593639890446680950 T^{4} - 853752814863508 p^{5} T^{5} + 13631111316 p^{10} T^{6} - 121100 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 144284 T + 13343871196 T^{2} - 729604203468780 T^{3} + 41427855222356636166 T^{4} - 729604203468780 p^{5} T^{5} + 13343871196 p^{10} T^{6} - 144284 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 6082 T + 4123518136 T^{2} - 80208632028694 T^{3} + 25918284294293674814 T^{4} - 80208632028694 p^{5} T^{5} + 4123518136 p^{10} T^{6} + 6082 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 43260 T + 17555442298 T^{2} - 587882039784852 T^{3} + \)\(13\!\cdots\!22\)\( T^{4} - 587882039784852 p^{5} T^{5} + 17555442298 p^{10} T^{6} - 43260 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 6862 T + 10323930298 T^{2} + 1651504026709662 T^{3} + 20416674410998484274 T^{4} + 1651504026709662 p^{5} T^{5} + 10323930298 p^{10} T^{6} + 6862 p^{15} T^{7} + p^{20} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.28536470246558265996210228849, −6.75703182645183434365380445574, −6.70066252978662387022882766953, −6.60013416826559938788894970292, −6.49991243858038671613734097158, −5.56916812235438299134443442831, −5.47980017227961197502989770805, −5.31228658379674816771639260819, −5.17419118967614531881823820514, −4.39456193803057284325743144110, −4.21660383978034999830046830743, −3.97711968628989544364247143225, −3.83926148696020834458429942717, −3.32947817198922500001471701669, −3.08680486680499968139836628227, −3.02893916777086754668929426466, −2.74224707688861817854190704537, −2.18580980149771089286769258993, −2.06563170752911657283740196605, −1.84295369798531709677678349983, −1.77852955548984555289461001209, −0.820846785257774734501779757734, −0.78702533463434135601803253129, −0.54346717931598525439916574571, −0.42807622797171266270546058512, 0.42807622797171266270546058512, 0.54346717931598525439916574571, 0.78702533463434135601803253129, 0.820846785257774734501779757734, 1.77852955548984555289461001209, 1.84295369798531709677678349983, 2.06563170752911657283740196605, 2.18580980149771089286769258993, 2.74224707688861817854190704537, 3.02893916777086754668929426466, 3.08680486680499968139836628227, 3.32947817198922500001471701669, 3.83926148696020834458429942717, 3.97711968628989544364247143225, 4.21660383978034999830046830743, 4.39456193803057284325743144110, 5.17419118967614531881823820514, 5.31228658379674816771639260819, 5.47980017227961197502989770805, 5.56916812235438299134443442831, 6.49991243858038671613734097158, 6.60013416826559938788894970292, 6.70066252978662387022882766953, 6.75703182645183434365380445574, 7.28536470246558265996210228849

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