Properties

Label 8-210e4-1.1-c3e4-0-4
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $23569.0$
Root an. cond. $3.52000$
Motivic weight $3$
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  + 4·2-s + 6·3-s + 4·4-s + 10·5-s + 24·6-s − 20·7-s − 16·8-s + 9·9-s + 40·10-s + 18·11-s + 24·12-s + 128·13-s − 80·14-s + 60·15-s − 64·16-s − 102·17-s + 36·18-s + 62·19-s + 40·20-s − 120·21-s + 72·22-s − 42·23-s − 96·24-s + 25·25-s + 512·26-s − 54·27-s − 80·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 1.07·7-s − 0.707·8-s + 1/3·9-s + 1.26·10-s + 0.493·11-s + 0.577·12-s + 2.73·13-s − 1.52·14-s + 1.03·15-s − 16-s − 1.45·17-s + 0.471·18-s + 0.748·19-s + 0.447·20-s − 1.24·21-s + 0.697·22-s − 0.380·23-s − 0.816·24-s + 1/5·25-s + 3.86·26-s − 0.384·27-s − 0.539·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(23569.0\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(14.34151311\)
\(L(\frac12)\) \(\approx\) \(14.34151311\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 20 T + 93 p T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
good11$D_4\times C_2$ \( 1 - 18 T - 1204 T^{2} + 20412 T^{3} + 131979 T^{4} + 20412 p^{3} T^{5} - 1204 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 64 T + 4203 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 6 p T - 808 T^{2} + 8316 p T^{3} + 50456523 T^{4} + 8316 p^{4} T^{5} - 808 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 62 T - 8675 T^{2} + 74338 T^{3} + 83313484 T^{4} + 74338 p^{3} T^{5} - 8675 p^{6} T^{6} - 62 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 42 T - 16396 T^{2} - 259308 T^{3} + 160287123 T^{4} - 259308 p^{3} T^{5} - 16396 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 222 T + 54484 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 50 T + 20053 T^{2} + 3856750 T^{3} - 653873372 T^{4} + 3856750 p^{3} T^{5} + 20053 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 572 T + 150697 T^{2} - 43003532 T^{3} + 12009083608 T^{4} - 43003532 p^{3} T^{5} + 150697 p^{6} T^{6} - 572 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 54 T + 137356 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 248 T + 171015 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 444 T - 57634 T^{2} + 20923056 T^{3} + 33661286643 T^{4} + 20923056 p^{3} T^{5} - 57634 p^{6} T^{6} + 444 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 12 T - 141586 T^{2} + 1872288 T^{3} - 2098406517 T^{4} + 1872288 p^{3} T^{5} - 141586 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 258 T - 87460 T^{2} - 66237372 T^{3} - 32127988821 T^{4} - 66237372 p^{3} T^{5} - 87460 p^{6} T^{6} + 258 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 64 T - 7250 p T^{2} - 487424 T^{3} + 146774663179 T^{4} - 487424 p^{3} T^{5} - 7250 p^{7} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 632 T - 117143 T^{2} + 53694088 T^{3} + 56251122808 T^{4} + 53694088 p^{3} T^{5} - 117143 p^{6} T^{6} - 632 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 78 T + 443968 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 464 T - 586187 T^{2} - 10880336 T^{3} + 407286968968 T^{4} - 10880336 p^{3} T^{5} - 586187 p^{6} T^{6} - 464 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 - 1946 T + 1856269 T^{2} - 1838131274 T^{3} + 1617213105364 T^{4} - 1838131274 p^{3} T^{5} + 1856269 p^{6} T^{6} - 1946 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 234 T + 181888 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 1674 T + 708104 T^{2} + 1145407716 T^{3} + 1913057365179 T^{4} + 1145407716 p^{3} T^{5} + 708104 p^{6} T^{6} + 1674 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 832 T + 1782402 T^{2} - 832 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.512717869820626408984389881979, −8.332779748308530954963856929002, −8.219180592439025218335279818125, −7.899960061065117581749789744864, −7.53196827517954980852188963306, −7.01373043977758972132349583124, −6.54904105253620163509834851718, −6.50338972284633379612834440254, −6.28004806575309218504272249814, −6.11424098561685173778580675136, −6.10823904864148618204778817458, −5.34658872254876761753605079223, −5.12142482764347127637446520601, −4.85091428285248487560898289389, −4.34212500033205738181498333907, −4.22515567297225110421096963391, −3.80606190393019456512081012823, −3.46470790548778947860713568262, −3.19641210301668849153500188750, −2.98626283765743198602633853329, −2.55112428346730330985558645896, −2.16266536538079544600407142675, −1.54974713627413221485302555176, −1.07968726238013717621673585583, −0.53583134612048889467506808526, 0.53583134612048889467506808526, 1.07968726238013717621673585583, 1.54974713627413221485302555176, 2.16266536538079544600407142675, 2.55112428346730330985558645896, 2.98626283765743198602633853329, 3.19641210301668849153500188750, 3.46470790548778947860713568262, 3.80606190393019456512081012823, 4.22515567297225110421096963391, 4.34212500033205738181498333907, 4.85091428285248487560898289389, 5.12142482764347127637446520601, 5.34658872254876761753605079223, 6.10823904864148618204778817458, 6.11424098561685173778580675136, 6.28004806575309218504272249814, 6.50338972284633379612834440254, 6.54904105253620163509834851718, 7.01373043977758972132349583124, 7.53196827517954980852188963306, 7.899960061065117581749789744864, 8.219180592439025218335279818125, 8.332779748308530954963856929002, 8.512717869820626408984389881979

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