Properties

Label 8-4032e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.643\times 10^{14}$
Sign $1$
Analytic cond. $1.07446\times 10^{6}$
Root an. cond. $5.67412$
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  − 20·25-s + 14·49-s + 16·67-s + 32·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·25-s + 2·49-s + 1.95·67-s + 3.60·79-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 + 4·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.07446\times 10^{6}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3395573019\)
\(L(\frac12)\) \(\approx\) \(0.3395573019\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^3$ \( 1 - 206 T^{4} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^3$ \( 1 - 734 T^{4} + p^{4} T^{8} \)
29$C_2^3$ \( 1 + 1234 T^{4} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^3$ \( 1 - 5582 T^{4} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
71$C_2^3$ \( 1 + 2914 T^{4} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−6.05444906419648009923081083954, −5.78649755878343244722395957762, −5.56523131212742323553356733252, −5.26550488290196437403607661252, −5.18919877161774478975524232421, −5.16110440468915807620758249581, −4.84983256668793828436023330567, −4.52011303949045395587626617689, −4.29752388814128003882379858894, −3.99092902762296418545644920640, −3.92226420067679130555353128367, −3.80651490657805057591021458403, −3.65518204889205029934487265029, −3.39843716138584351607481233628, −3.14254935113825269502291734549, −2.60324930068949114297693333655, −2.51505388792759405527698783706, −2.50263979703275888401011401198, −1.99565789840016076365660219284, −1.91894217027452077732245718210, −1.76028861041294364612386140118, −1.14203709059379675895118815030, −1.06514341566642674159890387925, −0.61058385485364190290755026385, −0.092270500147766648861724307336, 0.092270500147766648861724307336, 0.61058385485364190290755026385, 1.06514341566642674159890387925, 1.14203709059379675895118815030, 1.76028861041294364612386140118, 1.91894217027452077732245718210, 1.99565789840016076365660219284, 2.50263979703275888401011401198, 2.51505388792759405527698783706, 2.60324930068949114297693333655, 3.14254935113825269502291734549, 3.39843716138584351607481233628, 3.65518204889205029934487265029, 3.80651490657805057591021458403, 3.92226420067679130555353128367, 3.99092902762296418545644920640, 4.29752388814128003882379858894, 4.52011303949045395587626617689, 4.84983256668793828436023330567, 5.16110440468915807620758249581, 5.18919877161774478975524232421, 5.26550488290196437403607661252, 5.56523131212742323553356733252, 5.78649755878343244722395957762, 6.05444906419648009923081083954

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