Properties

Label 8-1280e4-1.1-c1e4-0-14
Degree $8$
Conductor $2.684\times 10^{12}$
Sign $1$
Analytic cond. $10913.1$
Root an. cond. $3.19700$
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  − 4·3-s + 8·9-s − 10·25-s − 20·27-s + 36·43-s + 12·67-s + 40·75-s + 50·81-s + 44·83-s + 24·89-s + 52·107-s + 44·121-s + 127-s − 144·129-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 + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s − 2·25-s − 3.84·27-s + 5.48·43-s + 1.46·67-s + 4.61·75-s + 50/9·81-s + 4.82·83-s + 2.54·89-s + 5.02·107-s + 4·121-s + 0.0887·127-s − 12.6·129-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10913.1\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.457780710\)
\(L(\frac12)\) \(\approx\) \(1.457780710\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
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_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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

−6.96323535993980912001408834720, −6.56501129617566401945718576998, −6.15152175551460638753145340114, −6.00411614949344167019037991834, −5.98366795293723770576581380274, −5.94927020370902942254644882972, −5.91632018248476031578471745370, −5.42167068402821246228966967119, −5.03345452595656004052874214796, −4.94917232199680345463683816598, −4.75840812015071141314183607097, −4.68744777663236629113160696692, −4.02388153881232411507310705479, −4.00863332317167002079233108185, −3.79899227438733933686813987936, −3.55037195662370158802562300755, −3.39572482943623920417532168379, −2.66543913310271005286843022162, −2.45782777237964079927509745783, −2.22142644494806211316697372181, −1.80807590031542927953216856909, −1.75874994722527744116461720097, −0.823694067411163150394965490612, −0.62899967962144220574872994114, −0.57622455115973410802870284720, 0.57622455115973410802870284720, 0.62899967962144220574872994114, 0.823694067411163150394965490612, 1.75874994722527744116461720097, 1.80807590031542927953216856909, 2.22142644494806211316697372181, 2.45782777237964079927509745783, 2.66543913310271005286843022162, 3.39572482943623920417532168379, 3.55037195662370158802562300755, 3.79899227438733933686813987936, 4.00863332317167002079233108185, 4.02388153881232411507310705479, 4.68744777663236629113160696692, 4.75840812015071141314183607097, 4.94917232199680345463683816598, 5.03345452595656004052874214796, 5.42167068402821246228966967119, 5.91632018248476031578471745370, 5.94927020370902942254644882972, 5.98366795293723770576581380274, 6.00411614949344167019037991834, 6.15152175551460638753145340114, 6.56501129617566401945718576998, 6.96323535993980912001408834720

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