Properties

Label 8-672e4-1.1-c1e4-0-11
Degree $8$
Conductor $203928109056$
Sign $1$
Analytic cond. $829.059$
Root an. cond. $2.31645$
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  + 2·9-s + 24·13-s + 16·25-s − 16·37-s − 2·49-s − 40·61-s + 56·73-s − 5·81-s + 24·97-s − 56·109-s + 48·117-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 308·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 6.65·13-s + 16/5·25-s − 2.63·37-s − 2/7·49-s − 5.12·61-s + 6.55·73-s − 5/9·81-s + 2.43·97-s − 5.36·109-s + 4.43·117-s − 3.63·121-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 + 23.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(829.059\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.738295757\)
\(L(\frac12)\) \(\approx\) \(5.738295757\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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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

−7.72142211051777158833357119158, −7.21214845940047040670319965215, −6.88586182618558396721729149147, −6.84176252730388801822811237722, −6.50573491882623869180484053423, −6.34670161889675517148478419690, −6.28178835773177346930465721014, −5.95323126076860277069743914398, −5.91068444137787962800747837318, −5.36982684689779848746050659130, −5.01028731405641212035465919608, −4.94589299982810611379148849924, −4.85658650311411509883963892841, −4.01493036803378932483330905584, −3.94810555666117317244170380124, −3.88572930058800319002859268860, −3.48844988906674948765317015983, −3.46180807729881421563979511984, −2.97707788277338935080946401619, −2.84537587019504311731275778316, −2.12091020995443491702709848618, −1.40371970220097143396361952811, −1.37957040085384193193361657961, −1.37006641816021559330365390279, −0.78275659100519606258338743859, 0.78275659100519606258338743859, 1.37006641816021559330365390279, 1.37957040085384193193361657961, 1.40371970220097143396361952811, 2.12091020995443491702709848618, 2.84537587019504311731275778316, 2.97707788277338935080946401619, 3.46180807729881421563979511984, 3.48844988906674948765317015983, 3.88572930058800319002859268860, 3.94810555666117317244170380124, 4.01493036803378932483330905584, 4.85658650311411509883963892841, 4.94589299982810611379148849924, 5.01028731405641212035465919608, 5.36982684689779848746050659130, 5.91068444137787962800747837318, 5.95323126076860277069743914398, 6.28178835773177346930465721014, 6.34670161889675517148478419690, 6.50573491882623869180484053423, 6.84176252730388801822811237722, 6.88586182618558396721729149147, 7.21214845940047040670319965215, 7.72142211051777158833357119158

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