Properties

Label 8-280e4-1.1-c1e4-0-6
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $24.9885$
Root an. cond. $1.49526$
Motivic weight $1$
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 + 4·3-s + 8·4-s − 4·5-s + 16·6-s + 8·8-s + 11·9-s − 16·10-s + 6·11-s + 32·12-s − 12·13-s − 16·15-s − 4·16-s + 6·17-s + 44·18-s + 12·19-s − 32·20-s + 24·22-s + 12·23-s + 32·24-s + 5·25-s − 48·26-s + 20·27-s − 4·29-s − 64·30-s − 18·31-s − 32·32-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 4·4-s − 1.78·5-s + 6.53·6-s + 2.82·8-s + 11/3·9-s − 5.05·10-s + 1.80·11-s + 9.23·12-s − 3.32·13-s − 4.13·15-s − 16-s + 1.45·17-s + 10.3·18-s + 2.75·19-s − 7.15·20-s + 5.11·22-s + 2.50·23-s + 6.53·24-s + 25-s − 9.41·26-s + 3.84·27-s − 0.742·29-s − 11.6·30-s − 3.23·31-s − 5.65·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(24.9885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.34834608\)
\(L(\frac12)\) \(\approx\) \(14.34834608\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$D_4\times C_2$ \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 6 T + 36 T^{2} - 144 T^{3} + 587 T^{4} - 144 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 45 T^{2} + 84 T^{3} - 1276 T^{4} + 84 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 36 T^{2} + 300 T^{3} + 551 T^{4} + 300 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 276 T^{3} + 4223 T^{4} - 276 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 18 T + 90 T^{2} + 528 T^{3} - 8377 T^{4} + 528 p T^{5} + 90 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 24 T + 180 T^{2} - 204 T^{3} - 9145 T^{4} - 204 p T^{5} + 180 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 24 T + 6 p T^{2} - 3888 T^{3} + 34091 T^{4} - 3888 p T^{5} + 6 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 12 T + 117 T^{2} + 996 T^{3} + 7052 T^{4} + 996 p T^{5} + 117 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 26 T + 290 T^{2} + 1776 T^{3} + 10223 T^{4} + 1776 p T^{5} + 290 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 24 T + 362 T^{2} - 4080 T^{3} + 37827 T^{4} - 4080 p T^{5} + 362 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 84 T^{3} - 4369 T^{4} - 84 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 24 T + 281 T^{2} - 2808 T^{3} + 27840 T^{4} - 2808 p T^{5} + 281 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 32 T + 512 T^{2} + 7008 T^{3} + 81038 T^{4} + 7008 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.856150741884632480976051535263, −8.013393780779249849632467863231, −7.985806684140267551315960356137, −7.74755342338894154378820966290, −7.41270832456188885269580035708, −7.34015116366196970148201708593, −7.18891416598227618562113383834, −6.91478729484676280178054894713, −6.74540192067330739695182476977, −6.29252902301538921845250620458, −5.63524496723586120461980288714, −5.44865415767917462709124295649, −5.10573423713970040928633239350, −5.06651400597055332186517501619, −4.78478021936815025531800260754, −4.35721761208681542601523186705, −3.90394222983684546146223926867, −3.82254502998517373614979056898, −3.51373938271878839443148989183, −3.38019176946268871953115662208, −3.21494422213599849990499540359, −2.68270785133712729551857699180, −2.30885854148395111787779706013, −1.98976639421233728469804954924, −1.07820900346388312666851896048, 1.07820900346388312666851896048, 1.98976639421233728469804954924, 2.30885854148395111787779706013, 2.68270785133712729551857699180, 3.21494422213599849990499540359, 3.38019176946268871953115662208, 3.51373938271878839443148989183, 3.82254502998517373614979056898, 3.90394222983684546146223926867, 4.35721761208681542601523186705, 4.78478021936815025531800260754, 5.06651400597055332186517501619, 5.10573423713970040928633239350, 5.44865415767917462709124295649, 5.63524496723586120461980288714, 6.29252902301538921845250620458, 6.74540192067330739695182476977, 6.91478729484676280178054894713, 7.18891416598227618562113383834, 7.34015116366196970148201708593, 7.41270832456188885269580035708, 7.74755342338894154378820966290, 7.985806684140267551315960356137, 8.013393780779249849632467863231, 8.856150741884632480976051535263

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