Properties

Degree $8$
Conductor $8.362\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·19-s − 4·25-s + 12·31-s + 16·37-s + 14·49-s − 60·103-s + 20·109-s + 20·121-s + 127-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 + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 5.50·19-s − 4/5·25-s + 2.15·31-s + 2.63·37-s + 2·49-s − 5.91·103-s + 1.91·109-s + 1.81·121-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 + 4·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.881245432\)
\(L(\frac12)\) \(\approx\) \(8.881245432\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 4 T^{2} - 9 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 20 T^{2} + 279 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 4 T^{2} - 513 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 - 68 T^{2} + 2943 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^3$ \( 1 + 100 T^{2} + 4959 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 + 172 T^{2} + 21663 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \)
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

−6.22899230271936802238522062419, −5.79266815760565350125540030818, −5.63053136825846376986787888824, −5.57640471948659405973464275448, −5.50528189964945061613132990178, −5.31735849532340681210473135984, −4.88292951646813613282425352156, −4.83482082952123882257171027579, −4.58811966439921181975386114843, −4.14742963629888266037061920740, −4.11975739136516895332749291846, −4.09808106434947960461563360132, −3.53534354442055969036632678668, −3.38065324834146286888349856525, −3.19257421683174542197157749086, −2.81721883695549176045427757706, −2.79298328559921511687295869755, −2.72740356856827842876243764385, −2.32283449726634844263267958265, −1.74909138116278568078884343740, −1.72525877072554997608860338842, −1.09233936792691387122644793982, −1.04163315293928335231985275052, −0.839469142595442730479484440327, −0.50976318418740276799249696482, 0.50976318418740276799249696482, 0.839469142595442730479484440327, 1.04163315293928335231985275052, 1.09233936792691387122644793982, 1.72525877072554997608860338842, 1.74909138116278568078884343740, 2.32283449726634844263267958265, 2.72740356856827842876243764385, 2.79298328559921511687295869755, 2.81721883695549176045427757706, 3.19257421683174542197157749086, 3.38065324834146286888349856525, 3.53534354442055969036632678668, 4.09808106434947960461563360132, 4.11975739136516895332749291846, 4.14742963629888266037061920740, 4.58811966439921181975386114843, 4.83482082952123882257171027579, 4.88292951646813613282425352156, 5.31735849532340681210473135984, 5.50528189964945061613132990178, 5.57640471948659405973464275448, 5.63053136825846376986787888824, 5.79266815760565350125540030818, 6.22899230271936802238522062419

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