Properties

Label 8-1183e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.959\times 10^{12}$
Sign $1$
Analytic cond. $7962.46$
Root an. cond. $3.07348$
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  + 4·4-s − 8·9-s + 4·16-s + 12·23-s − 2·25-s + 12·29-s − 32·36-s + 20·43-s − 2·49-s − 12·53-s + 24·61-s − 16·64-s + 28·79-s + 30·81-s + 48·92-s − 8·100-s − 24·101-s + 32·103-s − 24·107-s − 36·113-s + 48·116-s + 8·121-s + 127-s + 131-s + 137-s + 139-s − 32·144-s + ⋯
L(s)  = 1  + 2·4-s − 8/3·9-s + 16-s + 2.50·23-s − 2/5·25-s + 2.22·29-s − 5.33·36-s + 3.04·43-s − 2/7·49-s − 1.64·53-s + 3.07·61-s − 2·64-s + 3.15·79-s + 10/3·81-s + 5.00·92-s − 4/5·100-s − 2.38·101-s + 3.15·103-s − 2.32·107-s − 3.38·113-s + 4.45·116-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8/3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(7962.46\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.775077125\)
\(L(\frac12)\) \(\approx\) \(4.775077125\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( ( 1 + T^{2} )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
3$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^3$ \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 86 T^{2} + 3699 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 76 T^{2} + 3654 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 - 166 T^{2} + 11235 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 100 T^{2} + 4854 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 - 52 T^{2} - 714 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 112 T^{2} + 6018 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 206 T^{2} + 19467 T^{4} - 206 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 134 T^{2} + 12435 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 334 T^{2} + 43659 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 62 T^{2} + 19131 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.90849367139665123020828639816, −6.80593200785392358966371295315, −6.39768233294012560897383534203, −6.38809563866148815457226451948, −6.38162476015915415516721169592, −5.82586967476106075460732064030, −5.64853956109799753822711512629, −5.61711604944234924116434855354, −5.15702037904499284917634074172, −4.98004940742557123707613387922, −4.96868573668115840494075677370, −4.53016237893200899953251155818, −4.11483846106220186376339103959, −3.93484271803200795199379939235, −3.65361703743819101139162561290, −3.14887016743453032469639274128, −3.00611093591508705315434933700, −2.74061153298549914710427263892, −2.73768468888643032198507652173, −2.45600496581358883516590133258, −2.18203868392107879746775730904, −1.79555544560301329292161094314, −1.29139106747678635455856508547, −0.837244320492545906829699994381, −0.50298107073388121815470640618, 0.50298107073388121815470640618, 0.837244320492545906829699994381, 1.29139106747678635455856508547, 1.79555544560301329292161094314, 2.18203868392107879746775730904, 2.45600496581358883516590133258, 2.73768468888643032198507652173, 2.74061153298549914710427263892, 3.00611093591508705315434933700, 3.14887016743453032469639274128, 3.65361703743819101139162561290, 3.93484271803200795199379939235, 4.11483846106220186376339103959, 4.53016237893200899953251155818, 4.96868573668115840494075677370, 4.98004940742557123707613387922, 5.15702037904499284917634074172, 5.61711604944234924116434855354, 5.64853956109799753822711512629, 5.82586967476106075460732064030, 6.38162476015915415516721169592, 6.38809563866148815457226451948, 6.39768233294012560897383534203, 6.80593200785392358966371295315, 6.90849367139665123020828639816

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