Properties

Degree $8$
Conductor $3.263\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 8·25-s + 8·37-s + 16·43-s − 2·49-s + 32·67-s + 40·79-s − 9·81-s − 40·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1.51·7-s − 8/5·25-s + 1.31·37-s + 2.43·43-s − 2/7·49-s + 3.90·67-s + 4.50·79-s − 81-s − 3.83·109-s + 4·121-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 + 3.07·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.234161608\)
\(L(\frac12)\) \(\approx\) \(4.234161608\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.99570394146391909433856776171, −6.53973916320152757045221116717, −6.36062411835278405477238570711, −6.19284776678851533764778476548, −6.19015941168699676005615599687, −5.55127204461544351173617560034, −5.48528843572093334465524675341, −5.37598881321753188306594533393, −5.18130993612111320894790478933, −4.87588054465440716632789979036, −4.55927359551235072689699061888, −4.36665652472949880544494838885, −4.20186088531844883930035889134, −3.94434655622340833293609367367, −3.63810058301232834793265164490, −3.50211278522476471249498454669, −3.16954350753048597522444517988, −2.63687544521747585790303227821, −2.47948362807726104950042591516, −2.23815481024385545411522403883, −1.99038919907473091136245303705, −1.69131032866680340528989610033, −1.16417571717573110167605445649, −0.929404761008709542750864254335, −0.45219101148943517851934378369, 0.45219101148943517851934378369, 0.929404761008709542750864254335, 1.16417571717573110167605445649, 1.69131032866680340528989610033, 1.99038919907473091136245303705, 2.23815481024385545411522403883, 2.47948362807726104950042591516, 2.63687544521747585790303227821, 3.16954350753048597522444517988, 3.50211278522476471249498454669, 3.63810058301232834793265164490, 3.94434655622340833293609367367, 4.20186088531844883930035889134, 4.36665652472949880544494838885, 4.55927359551235072689699061888, 4.87588054465440716632789979036, 5.18130993612111320894790478933, 5.37598881321753188306594533393, 5.48528843572093334465524675341, 5.55127204461544351173617560034, 6.19015941168699676005615599687, 6.19284776678851533764778476548, 6.36062411835278405477238570711, 6.53973916320152757045221116717, 6.99570394146391909433856776171

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