Properties

Label 8-7728e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.567\times 10^{15}$
Sign $1$
Analytic cond. $1.45002\times 10^{7}$
Root an. cond. $7.85546$
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  − 4·3-s − 2·5-s + 4·7-s + 10·9-s + 2·13-s + 8·15-s − 10·17-s − 4·19-s − 16·21-s + 4·23-s − 8·25-s − 20·27-s + 11·29-s − 14·31-s − 8·35-s + 13·37-s − 8·39-s + 7·41-s − 12·43-s − 20·45-s − 15·47-s + 10·49-s + 40·51-s + 53-s + 16·57-s − 5·59-s + 3·61-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 1.51·7-s + 10/3·9-s + 0.554·13-s + 2.06·15-s − 2.42·17-s − 0.917·19-s − 3.49·21-s + 0.834·23-s − 8/5·25-s − 3.84·27-s + 2.04·29-s − 2.51·31-s − 1.35·35-s + 2.13·37-s − 1.28·39-s + 1.09·41-s − 1.82·43-s − 2.98·45-s − 2.18·47-s + 10/7·49-s + 5.60·51-s + 0.137·53-s + 2.11·57-s − 0.650·59-s + 0.384·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.45002\times 10^{7}\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 12 T^{2} + 21 T^{3} + 68 T^{4} + 21 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 7 T^{2} + 232 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 24 T^{2} - 49 T^{3} + 474 T^{4} - 49 p T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 69 T^{2} + 206 T^{3} + 1908 T^{4} + 206 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 121 T^{2} - 718 T^{3} + 4782 T^{4} - 718 p T^{5} + 121 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 197 T^{2} - 1498 T^{3} + 11850 T^{4} - 1498 p T^{5} + 197 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 125 T^{2} - 626 T^{3} + 6898 T^{4} - 626 p T^{5} + 125 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 134 T^{2} + 659 T^{3} + 5190 T^{4} + 659 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 232 T^{2} + 1923 T^{3} + 16614 T^{4} + 1923 p T^{5} + 232 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - T + 185 T^{2} - 201 T^{3} + 13976 T^{4} - 201 p T^{5} + 185 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 129 T^{2} + 435 T^{3} + 9596 T^{4} + 435 p T^{5} + 129 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 17 T^{2} + 251 T^{3} + 2652 T^{4} + 251 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 257 T^{2} + 973 T^{3} + 25444 T^{4} + 973 p T^{5} + 257 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 165 T^{2} + 1119 T^{3} + 14732 T^{4} + 1119 p T^{5} + 165 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 199 T^{2} + 652 T^{3} + 17348 T^{4} + 652 p T^{5} + 199 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 317 T^{2} + 2554 T^{3} + 20452 T^{4} + 2554 p T^{5} + 317 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 171 T^{2} - 20 T^{3} + 15384 T^{4} - 20 p T^{5} + 171 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 285 T^{2} - 1255 T^{3} + 34016 T^{4} - 1255 p T^{5} + 285 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 27 T + 491 T^{2} + 6242 T^{3} + 69216 T^{4} + 6242 p T^{5} + 491 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.80368075537589679291156329260, −5.59559766264292863307994073350, −5.48790034776395795929884703886, −5.45835838900808342375065057205, −5.19973263868615367513059241316, −4.82365644963971572518801292763, −4.72007448947647001786061271003, −4.67329642749081535122339221109, −4.46394387604214902351691251387, −4.22982228330512844970581926760, −4.14626059388879338457055550121, −4.12385328341700861509586040963, −3.85420856909961788580721791298, −3.45597522361879496009887486711, −3.29232301597384254593855214515, −3.01323682147079638040503482252, −2.90790531659365636493665943365, −2.28960575433093441139367157009, −2.21188199843812964952850075081, −2.15793566670971130851909772768, −1.93994389665512026441717492821, −1.36787069694414502873132455750, −1.31847790525187226101400965227, −1.13480403969365400821955902352, −1.00308738090636194998451174759, 0, 0, 0, 0, 1.00308738090636194998451174759, 1.13480403969365400821955902352, 1.31847790525187226101400965227, 1.36787069694414502873132455750, 1.93994389665512026441717492821, 2.15793566670971130851909772768, 2.21188199843812964952850075081, 2.28960575433093441139367157009, 2.90790531659365636493665943365, 3.01323682147079638040503482252, 3.29232301597384254593855214515, 3.45597522361879496009887486711, 3.85420856909961788580721791298, 4.12385328341700861509586040963, 4.14626059388879338457055550121, 4.22982228330512844970581926760, 4.46394387604214902351691251387, 4.67329642749081535122339221109, 4.72007448947647001786061271003, 4.82365644963971572518801292763, 5.19973263868615367513059241316, 5.45835838900808342375065057205, 5.48790034776395795929884703886, 5.59559766264292863307994073350, 5.80368075537589679291156329260

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