Properties

Label 8-4032e4-1.1-c1e4-0-22
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  + 8·13-s + 4·25-s − 2·49-s + 56·61-s − 24·73-s − 56·97-s − 48·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2.21·13-s + 4/5·25-s − 2/7·49-s + 7.17·61-s − 2.80·73-s − 5.68·97-s − 4.59·109-s − 3.63·121-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 − 0.923·169-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 + ⋯

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

\(L(1)\) \(\approx\) \(5.189140664\)
\(L(\frac12)\) \(\approx\) \(5.189140664\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
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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

−5.90379231733636370091699555808, −5.79646367380113618093251967133, −5.75953146419699791890465920815, −5.24309889351418067578033301150, −5.17021160182888472689657312128, −5.09137582391419915812207194204, −4.97208474739099251564042827583, −4.45574951187842142800938981504, −4.29981104990853312764551742033, −3.99731226791336984636867128295, −3.85483962267815838752224897382, −3.82659889905347132719623806491, −3.69654397458359394624669619492, −3.40291797692853083245820308844, −3.05517010390007903827816977894, −2.61775185611172920938863115321, −2.58189510653713276519139774908, −2.45778872708684952647701927580, −2.39700734866902986619606575740, −1.54272228800375929072934093144, −1.39134891920474561152398552055, −1.31391433769813570365900192226, −1.29826683128971216750957846033, −0.50418021601403730046213203027, −0.41360937206465012608049335155, 0.41360937206465012608049335155, 0.50418021601403730046213203027, 1.29826683128971216750957846033, 1.31391433769813570365900192226, 1.39134891920474561152398552055, 1.54272228800375929072934093144, 2.39700734866902986619606575740, 2.45778872708684952647701927580, 2.58189510653713276519139774908, 2.61775185611172920938863115321, 3.05517010390007903827816977894, 3.40291797692853083245820308844, 3.69654397458359394624669619492, 3.82659889905347132719623806491, 3.85483962267815838752224897382, 3.99731226791336984636867128295, 4.29981104990853312764551742033, 4.45574951187842142800938981504, 4.97208474739099251564042827583, 5.09137582391419915812207194204, 5.17021160182888472689657312128, 5.24309889351418067578033301150, 5.75953146419699791890465920815, 5.79646367380113618093251967133, 5.90379231733636370091699555808

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