Properties

Label 8-2646e4-1.1-c1e4-0-11
Degree $8$
Conductor $4.902\times 10^{13}$
Sign $1$
Analytic cond. $199281.$
Root an. cond. $4.59656$
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  − 2·4-s + 3·16-s − 14·25-s + 16·37-s − 32·43-s − 4·64-s + 56·67-s + 44·79-s + 28·100-s + 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 64·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 3/4·16-s − 2.79·25-s + 2.63·37-s − 4.87·43-s − 1/2·64-s + 6.84·67-s + 4.95·79-s + 14/5·100-s + 6.13·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 4.87·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(199281.\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2646} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.114992769\)
\(L(\frac12)\) \(\approx\) \(3.114992769\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
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

−6.45767220955065700803867821336, −5.97583118186540710589804657705, −5.84785856360911156112255799902, −5.76138974144281889441495764637, −5.30455612163189332285704815784, −5.16421543971665505428944855316, −5.08103161813998801367509538082, −4.84092697095315774031840891271, −4.82330606447014498129020397812, −4.31862279008079318451492680646, −4.10140459745278632536157767856, −4.01788752085104880234932400964, −3.66782316406694439840228095954, −3.63357576461052792250716279637, −3.24680385189570790314893199631, −3.23635022536193382397214840423, −2.92452581889107927982024930614, −2.28855132632752118651242972801, −2.09915777780361480466238075022, −2.01981427217287942549619939831, −1.95847137352763375268975044676, −1.39985817338617933877792908582, −0.72160017936998544412779025253, −0.71391549139096931512885407112, −0.43016750187602492803513983418, 0.43016750187602492803513983418, 0.71391549139096931512885407112, 0.72160017936998544412779025253, 1.39985817338617933877792908582, 1.95847137352763375268975044676, 2.01981427217287942549619939831, 2.09915777780361480466238075022, 2.28855132632752118651242972801, 2.92452581889107927982024930614, 3.23635022536193382397214840423, 3.24680385189570790314893199631, 3.63357576461052792250716279637, 3.66782316406694439840228095954, 4.01788752085104880234932400964, 4.10140459745278632536157767856, 4.31862279008079318451492680646, 4.82330606447014498129020397812, 4.84092697095315774031840891271, 5.08103161813998801367509538082, 5.16421543971665505428944855316, 5.30455612163189332285704815784, 5.76138974144281889441495764637, 5.84785856360911156112255799902, 5.97583118186540710589804657705, 6.45767220955065700803867821336

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