Properties

Label 8-384e4-1.1-c1e4-0-6
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $88.3961$
Root an. cond. $1.75107$
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  − 2·9-s − 8·17-s + 4·25-s + 40·41-s − 12·49-s − 8·73-s + 3·81-s + 24·89-s + 56·97-s + 8·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s − 1.94·17-s + 4/5·25-s + 6.24·41-s − 1.71·49-s − 0.936·73-s + 1/3·81-s + 2.54·89-s + 5.68·97-s + 0.752·113-s + 1.09·121-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 + 1.29·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(88.3961\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.849726689\)
\(L(\frac12)\) \(\approx\) \(1.849726689\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
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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

−8.049838974768667062494455871664, −7.955343462929157905281491771570, −7.78189179184266387615888390757, −7.51160963545188632611326604934, −7.31227510997071894143389406489, −6.94803412047455459878481713307, −6.73331959112034251565986066026, −6.26814282724700982845983465206, −6.26220373076718752600619059946, −5.89665677343569916421534970387, −5.89445475670083262665834723464, −5.42757163728111299870223051784, −5.05460623651132301736697090399, −4.59014808961757648683749422747, −4.51608294301034087429074745230, −4.46856019048839960304531083966, −4.01576280278593421328541002766, −3.52284000970033550234053273583, −3.19584874368916845285221376212, −3.01250857646344909223200469975, −2.40438796828212904038147726604, −2.12688548092163566730867907666, −2.11078420427884692762323138139, −1.02510739554723904828571736839, −0.63161969456271573036383893493, 0.63161969456271573036383893493, 1.02510739554723904828571736839, 2.11078420427884692762323138139, 2.12688548092163566730867907666, 2.40438796828212904038147726604, 3.01250857646344909223200469975, 3.19584874368916845285221376212, 3.52284000970033550234053273583, 4.01576280278593421328541002766, 4.46856019048839960304531083966, 4.51608294301034087429074745230, 4.59014808961757648683749422747, 5.05460623651132301736697090399, 5.42757163728111299870223051784, 5.89445475670083262665834723464, 5.89665677343569916421534970387, 6.26220373076718752600619059946, 6.26814282724700982845983465206, 6.73331959112034251565986066026, 6.94803412047455459878481713307, 7.31227510997071894143389406489, 7.51160963545188632611326604934, 7.78189179184266387615888390757, 7.955343462929157905281491771570, 8.049838974768667062494455871664

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