Properties

Label 8-40e8-1.1-c1e4-0-4
Degree $8$
Conductor $6.554\times 10^{12}$
Sign $1$
Analytic cond. $26643.3$
Root an. cond. $3.57436$
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 − 2·9-s + 40·27-s − 36·41-s + 16·43-s + 4·49-s + 44·67-s − 55·81-s − 60·83-s + 12·89-s − 36·107-s + 26·121-s + 144·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s − 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2.30·3-s − 2/3·9-s + 7.69·27-s − 5.62·41-s + 2.43·43-s + 4/7·49-s + 5.37·67-s − 6.11·81-s − 6.58·83-s + 1.27·89-s − 3.48·107-s + 2.36·121-s + 12.9·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(26643.3\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2705440488\)
\(L(\frac12)\) \(\approx\) \(0.2705440488\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.61225455370843592692080664224, −6.45315831269802301340573313322, −6.25440215184288638670466447591, −5.95509094023615781303372443677, −5.81611767167689992964189465276, −5.64575826171086308817580322910, −5.36111100221311183335086572718, −5.25729243646382768443004688434, −5.09465194984217825982339193373, −5.02877852907309588390505642347, −4.64422844565878597205287770010, −4.48957355605712433354969311397, −3.92632188978250475156374063359, −3.77067883993671132444663164287, −3.67396254512510274084575324030, −3.22777628827437212835551423331, −2.98292513368384897572339072330, −2.73748813385637224814351707850, −2.44521571234295434670274470986, −2.39152664843649876518415424324, −1.71852454456417866311494273298, −1.43170118626755737657459609348, −0.982186635103503665799593854644, −0.48810051878648358844757487152, −0.23276194127581505080245378699, 0.23276194127581505080245378699, 0.48810051878648358844757487152, 0.982186635103503665799593854644, 1.43170118626755737657459609348, 1.71852454456417866311494273298, 2.39152664843649876518415424324, 2.44521571234295434670274470986, 2.73748813385637224814351707850, 2.98292513368384897572339072330, 3.22777628827437212835551423331, 3.67396254512510274084575324030, 3.77067883993671132444663164287, 3.92632188978250475156374063359, 4.48957355605712433354969311397, 4.64422844565878597205287770010, 5.02877852907309588390505642347, 5.09465194984217825982339193373, 5.25729243646382768443004688434, 5.36111100221311183335086572718, 5.64575826171086308817580322910, 5.81611767167689992964189465276, 5.95509094023615781303372443677, 6.25440215184288638670466447591, 6.45315831269802301340573313322, 6.61225455370843592692080664224

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