Properties

Label 8-384e4-1.1-c4e4-0-3
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.48257\times 10^{6}$
Root an. cond. $6.30032$
Motivic weight $4$
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  − 12·3-s − 54·9-s + 536·11-s + 380·25-s + 2.05e3·27-s − 6.43e3·33-s − 6.72e3·49-s − 5.52e3·59-s + 1.95e4·73-s − 4.56e3·75-s − 8.82e3·81-s − 2.36e4·83-s + 2.34e4·97-s − 2.89e4·99-s + 8.50e4·107-s + 1.20e5·121-s + 127-s + 131-s + 137-s + 139-s + 8.06e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.31e4·169-s + ⋯
L(s)  = 1  − 4/3·3-s − 2/3·9-s + 4.42·11-s + 0.607·25-s + 2.81·27-s − 5.90·33-s − 2.80·49-s − 1.58·59-s + 3.67·73-s − 0.810·75-s − 1.34·81-s − 3.43·83-s + 2.49·97-s − 2.95·99-s + 7.43·107-s + 8.26·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.73·147-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.26·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.48257\times 10^{6}\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.121363472\)
\(L(\frac12)\) \(\approx\) \(4.121363472\)
\(L(3)\) 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$C_2$ \( ( 1 + 2 p T + p^{4} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 38 p T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 3362 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 134 T + p^{4} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + 46558 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 110594 T^{2} + p^{8} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 18434 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 144962 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 779522 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 636002 T^{2} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 1156322 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3988034 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 6657602 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3123842 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 13118402 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 1382 T + p^{4} T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 27588002 T^{2} + p^{8} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 40267394 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 47090882 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4894 T + p^{4} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 75709922 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2$ \( ( 1 + 5914 T + p^{4} T^{2} )^{4} \)
89$C_2$ \( ( 1 - 146 p T + p^{4} T^{2} )^{2}( 1 + 146 p T + p^{4} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5858 T + p^{4} T^{2} )^{4} \)
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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

−7.42971798895615412475464463932, −7.16704250306212038308531364638, −6.94036016089278765614015042094, −6.58591718895229054336555355856, −6.55683486557240778684909366046, −6.24908175476181567170742756798, −6.01132966089653162991311009247, −6.00781262385241006909935091129, −5.75443877622999579663147582544, −5.07849028101878372972399635501, −4.89334956767050092290594417300, −4.83394479051005586261730082593, −4.37086732226604711774496694337, −4.27962885669750693861051280121, −3.64394360595061051881950356873, −3.55162809193684666359752005690, −3.32660931717580444725000056803, −3.05906629281221304184114089871, −2.48426995633550173275920915321, −1.88333305217620891750171785459, −1.76862549458342717227578103905, −1.30131357889534417322863241613, −0.985631929268675414358453245315, −0.61970365470539023692694636437, −0.39022030855187228989397258222, 0.39022030855187228989397258222, 0.61970365470539023692694636437, 0.985631929268675414358453245315, 1.30131357889534417322863241613, 1.76862549458342717227578103905, 1.88333305217620891750171785459, 2.48426995633550173275920915321, 3.05906629281221304184114089871, 3.32660931717580444725000056803, 3.55162809193684666359752005690, 3.64394360595061051881950356873, 4.27962885669750693861051280121, 4.37086732226604711774496694337, 4.83394479051005586261730082593, 4.89334956767050092290594417300, 5.07849028101878372972399635501, 5.75443877622999579663147582544, 6.00781262385241006909935091129, 6.01132966089653162991311009247, 6.24908175476181567170742756798, 6.55683486557240778684909366046, 6.58591718895229054336555355856, 6.94036016089278765614015042094, 7.16704250306212038308531364638, 7.42971798895615412475464463932

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