Properties

Label 8-882e4-1.1-c4e4-0-1
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $6.90958\times 10^{7}$
Root an. cond. $9.54841$
Motivic weight $4$
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  − 16·4-s + 192·16-s + 804·25-s + 6.24e3·37-s + 1.31e3·43-s − 2.04e3·64-s − 4.52e3·67-s + 3.46e4·79-s − 1.28e4·100-s − 3.90e4·109-s + 5.85e4·121-s + 127-s + 131-s + 137-s + 139-s − 9.98e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.00e5·169-s − 2.09e4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 3/4·16-s + 1.28·25-s + 4.55·37-s + 0.709·43-s − 1/2·64-s − 1.00·67-s + 5.54·79-s − 1.28·100-s − 3.28·109-s + 3.99·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 4.55·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.52·169-s − 0.709·172-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.90958\times 10^{7}\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6450674857\)
\(L(\frac12)\) \(\approx\) \(0.6450674857\)
\(L(3)\) 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$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 402 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 29280 T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 50338 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 98354 T^{2} + p^{8} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 55426 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 555632 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1162512 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 226 p^{2} T^{2} + p^{8} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 1560 T + p^{4} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 3934322 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 328 T + p^{4} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 7339710 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 316176 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 23661474 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 26265346 T^{2} + p^{8} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 1130 T + p^{4} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 50644560 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 8962 p^{2} T^{2} + p^{8} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 8650 T + p^{4} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 86094050 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 51619470 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 4471906 T^{2} + p^{8} T^{4} )^{2} \)
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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

−6.60519326297852851946862027769, −6.46145331616278844822113290058, −6.34686895478052535422414520281, −5.93945147705536932620912619047, −5.78118175206049894774872281457, −5.46409291015710883613888634335, −5.33509911349211368360540050065, −4.82275428474841334664954044370, −4.76472783879323332505856011094, −4.63372325112143852873865490616, −4.51599013269403022076640406116, −3.93344531914360736715766275115, −3.73030339672278722979417149009, −3.59533640703128959160894617004, −3.58608231993250748045042434184, −2.64896769813970779775048885002, −2.64805040023020560205062173458, −2.64399587532939905400777985427, −2.36409488237303216540581207121, −1.74497786122304326811342273651, −1.28878421588695916333396871625, −1.06516604928504321879807441212, −0.946339761930522299084069044958, −0.59314186783584311001533297087, −0.096675206597797361651727388960, 0.096675206597797361651727388960, 0.59314186783584311001533297087, 0.946339761930522299084069044958, 1.06516604928504321879807441212, 1.28878421588695916333396871625, 1.74497786122304326811342273651, 2.36409488237303216540581207121, 2.64399587532939905400777985427, 2.64805040023020560205062173458, 2.64896769813970779775048885002, 3.58608231993250748045042434184, 3.59533640703128959160894617004, 3.73030339672278722979417149009, 3.93344531914360736715766275115, 4.51599013269403022076640406116, 4.63372325112143852873865490616, 4.76472783879323332505856011094, 4.82275428474841334664954044370, 5.33509911349211368360540050065, 5.46409291015710883613888634335, 5.78118175206049894774872281457, 5.93945147705536932620912619047, 6.34686895478052535422414520281, 6.46145331616278844822113290058, 6.60519326297852851946862027769

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