Properties

Label 8-3024e4-1.1-c1e4-0-2
Degree $8$
Conductor $8.362\times 10^{13}$
Sign $1$
Analytic cond. $339966.$
Root an. cond. $4.91393$
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  − 8·7-s + 4·25-s − 8·37-s + 44·43-s + 34·49-s + 28·67-s − 32·79-s + 16·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 3.02·7-s + 4/5·25-s − 1.31·37-s + 6.70·43-s + 34/7·49-s + 3.42·67-s − 3.60·79-s + 1.53·109-s − 2.54·121-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 + 3.53·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(339966.\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5168178586\)
\(L(\frac12)\) \(\approx\) \(0.5168178586\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.20480968573751853862534011159, −6.06107382667964007208527650944, −5.84419731698912348304469408878, −5.63882476350076115983387494833, −5.39318161614399163435073121046, −5.24491206070857153428729584453, −5.11427556410438191973106388713, −4.76193153964368895152295759134, −4.34127798906834225057164771624, −4.12302560734193529053718358221, −4.05983552332924558221594504943, −3.93785290566110652860033438457, −3.79108577758073422401579256900, −3.38197096344266498248825187001, −3.05919087698111679810180064815, −2.97237657547156576133852766029, −2.81970290182233458631684285037, −2.56708568040220006853202737918, −2.23905039686556403715569207046, −2.21641373888780955690045618647, −1.66021960966188214002536986596, −1.15342635784500133827281101525, −0.864449789821986888192697866052, −0.74721693379687147814401437040, −0.13955573195696600379917867544, 0.13955573195696600379917867544, 0.74721693379687147814401437040, 0.864449789821986888192697866052, 1.15342635784500133827281101525, 1.66021960966188214002536986596, 2.21641373888780955690045618647, 2.23905039686556403715569207046, 2.56708568040220006853202737918, 2.81970290182233458631684285037, 2.97237657547156576133852766029, 3.05919087698111679810180064815, 3.38197096344266498248825187001, 3.79108577758073422401579256900, 3.93785290566110652860033438457, 4.05983552332924558221594504943, 4.12302560734193529053718358221, 4.34127798906834225057164771624, 4.76193153964368895152295759134, 5.11427556410438191973106388713, 5.24491206070857153428729584453, 5.39318161614399163435073121046, 5.63882476350076115983387494833, 5.84419731698912348304469408878, 6.06107382667964007208527650944, 6.20480968573751853862534011159

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