Properties

Label 8-280e4-1.1-c1e4-0-10
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 $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 2·3-s + 8·4-s − 4·5-s + 8·6-s − 10·7-s − 8·8-s − 9-s + 16·10-s + 6·11-s − 16·12-s − 12·13-s + 40·14-s + 8·15-s − 4·16-s − 6·17-s + 4·18-s − 12·19-s − 32·20-s + 20·21-s − 24·22-s − 6·23-s + 16·24-s + 5·25-s + 48·26-s + 2·27-s − 80·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 4·4-s − 1.78·5-s + 3.26·6-s − 3.77·7-s − 2.82·8-s − 1/3·9-s + 5.05·10-s + 1.80·11-s − 4.61·12-s − 3.32·13-s + 10.6·14-s + 2.06·15-s − 16-s − 1.45·17-s + 0.942·18-s − 2.75·19-s − 7.15·20-s + 4.36·21-s − 5.11·22-s − 1.25·23-s + 3.26·24-s + 25-s + 9.41·26-s + 0.384·27-s − 15.1·28-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: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
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$ \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$D_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 246 T^{3} + 1208 T^{4} + 246 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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$D_4\times C_2$ \( 1 + 12 T + 36 T^{2} - 12 p T^{3} - 4777 T^{4} - 12 p^{2} T^{5} + 36 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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$D_4\times C_2$ \( 1 - 6 T + 90 T^{2} - 672 T^{3} + 5159 T^{4} - 672 p T^{5} + 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 12 T + 180 T^{2} + 1500 T^{3} + 13751 T^{4} + 1500 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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$D_4\times C_2$ \( 1 + 18 T + 117 T^{2} - 6 T^{3} - 5008 T^{4} - 6 p T^{5} + 117 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 22 T + 290 T^{2} + 2736 T^{3} + 24239 T^{4} + 2736 p T^{5} + 290 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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

−9.198517183680155927992841623590, −8.895628573013395198214926459934, −8.809742612845560140779735292210, −8.638380396110058048965093890751, −8.294107361977494219496444995427, −7.67743553580030210321353504033, −7.60962458049685940924292790643, −7.32498470385754554009418538853, −7.31159751016880320366546950420, −6.81159856042642961845173788890, −6.64746783215604572579625498997, −6.55945260221215045398144786333, −6.27963831145766162924715230676, −6.08168498421144830718976732131, −5.67546771380448751773039459399, −4.97854142869800318530993248435, −4.91998841012592286652111694114, −4.29332914125950315377178814389, −4.13221213250444392625625761669, −3.76899561126095760582417579529, −3.64600100804541746265500342734, −2.94376085234748517950206501713, −2.59475920710829249919940461625, −2.11489822527511908787053687211, −1.80849615249347930090697781753, 0, 0, 0, 0, 1.80849615249347930090697781753, 2.11489822527511908787053687211, 2.59475920710829249919940461625, 2.94376085234748517950206501713, 3.64600100804541746265500342734, 3.76899561126095760582417579529, 4.13221213250444392625625761669, 4.29332914125950315377178814389, 4.91998841012592286652111694114, 4.97854142869800318530993248435, 5.67546771380448751773039459399, 6.08168498421144830718976732131, 6.27963831145766162924715230676, 6.55945260221215045398144786333, 6.64746783215604572579625498997, 6.81159856042642961845173788890, 7.31159751016880320366546950420, 7.32498470385754554009418538853, 7.60962458049685940924292790643, 7.67743553580030210321353504033, 8.294107361977494219496444995427, 8.638380396110058048965093890751, 8.809742612845560140779735292210, 8.895628573013395198214926459934, 9.198517183680155927992841623590

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