Properties

Label 8-2100e4-1.1-c1e4-0-15
Degree $8$
Conductor $1.945\times 10^{13}$
Sign $1$
Analytic cond. $79065.2$
Root an. cond. $4.09494$
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  + 9-s − 4·11-s + 8·19-s − 16·29-s − 6·31-s + 24·41-s − 2·49-s − 12·59-s + 2·61-s + 40·71-s + 14·79-s + 20·89-s − 4·99-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 8·171-s + 173-s + ⋯
L(s)  = 1  + 1/3·9-s − 1.20·11-s + 1.83·19-s − 2.97·29-s − 1.07·31-s + 3.74·41-s − 2/7·49-s − 1.56·59-s + 0.256·61-s + 4.74·71-s + 1.57·79-s + 2.11·89-s − 0.402·99-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.611·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(79065.2\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.188393093\)
\(L(\frac12)\) \(\approx\) \(4.188393093\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 - 7 T^{2} - 1320 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 102 T^{2} + 7595 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 145 T^{2} + 15696 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.37032846085215038524367926285, −6.13279768253071941751462265991, −5.97514152426270705443206661700, −5.90289226258669110588488882676, −5.61097645566576841973699276621, −5.47799239380872241279304490081, −5.08028752522182629966220178152, −5.07580866531011820359701447742, −4.92437796143914723075494856445, −4.62331873963495968221215442833, −4.18748422911625635747606038361, −4.16327220724221357259013396289, −3.82243157560705699673072942615, −3.54039929745143860598433886378, −3.43650300761406024814772363261, −3.14627610221115201403591347665, −3.05588597465297942748305154182, −2.47824314985217699343970059238, −2.22747264651896500106427872191, −2.18915391557543647366357560437, −1.87068492828571994887887238356, −1.55706665109378595730675548313, −0.954915934969624229202126696875, −0.68572778958977538186568091797, −0.46848921793548418114768874206, 0.46848921793548418114768874206, 0.68572778958977538186568091797, 0.954915934969624229202126696875, 1.55706665109378595730675548313, 1.87068492828571994887887238356, 2.18915391557543647366357560437, 2.22747264651896500106427872191, 2.47824314985217699343970059238, 3.05588597465297942748305154182, 3.14627610221115201403591347665, 3.43650300761406024814772363261, 3.54039929745143860598433886378, 3.82243157560705699673072942615, 4.16327220724221357259013396289, 4.18748422911625635747606038361, 4.62331873963495968221215442833, 4.92437796143914723075494856445, 5.07580866531011820359701447742, 5.08028752522182629966220178152, 5.47799239380872241279304490081, 5.61097645566576841973699276621, 5.90289226258669110588488882676, 5.97514152426270705443206661700, 6.13279768253071941751462265991, 6.37032846085215038524367926285

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