Properties

Label 8-2352e4-1.1-c1e4-0-11
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·9-s − 16·13-s − 2·25-s − 4·37-s + 12·61-s + 28·73-s + 16·81-s + 32·97-s + 20·109-s − 80·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 5/3·9-s − 4.43·13-s − 2/5·25-s − 0.657·37-s + 1.53·61-s + 3.27·73-s + 16/9·81-s + 3.24·97-s + 1.91·109-s − 7.39·117-s − 2·121-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 + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.018619873\)
\(L(\frac12)\) \(\approx\) \(3.018619873\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 167 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
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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

−6.40002027221073859349949252581, −6.38130678507351713748243794154, −6.04841372145232596017626697262, −5.59953796382860905098672878526, −5.36370043080175001735708744818, −5.20264489347812509052820192454, −5.11300003502230271010769247357, −4.99688988708014104975088302908, −4.80404672906441903959002871156, −4.37495988091423507285528000430, −4.29162284538368623740324640065, −4.25812783298571084350987260458, −3.71973113080277983151233657025, −3.67325216289455862058976769645, −3.38123783756840564763076096595, −2.97252362417964098296084082983, −2.72267718293821662937868102610, −2.59473417873849938688599457430, −2.22570325336136179062630474824, −2.02447006964472145258521515744, −1.77135205231088166297709692721, −1.74781712990058556735330510964, −0.929981671462054324155053037832, −0.59744382209633611690000424288, −0.40281135957156093806881923988, 0.40281135957156093806881923988, 0.59744382209633611690000424288, 0.929981671462054324155053037832, 1.74781712990058556735330510964, 1.77135205231088166297709692721, 2.02447006964472145258521515744, 2.22570325336136179062630474824, 2.59473417873849938688599457430, 2.72267718293821662937868102610, 2.97252362417964098296084082983, 3.38123783756840564763076096595, 3.67325216289455862058976769645, 3.71973113080277983151233657025, 4.25812783298571084350987260458, 4.29162284538368623740324640065, 4.37495988091423507285528000430, 4.80404672906441903959002871156, 4.99688988708014104975088302908, 5.11300003502230271010769247357, 5.20264489347812509052820192454, 5.36370043080175001735708744818, 5.59953796382860905098672878526, 6.04841372145232596017626697262, 6.38130678507351713748243794154, 6.40002027221073859349949252581

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