Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{12} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s + 10·25-s − 4·37-s − 8·43-s + 34·49-s + 40·67-s − 8·79-s − 28·109-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 80·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 3.02·7-s + 2·25-s − 0.657·37-s − 1.21·43-s + 34/7·49-s + 4.88·67-s − 0.900·79-s − 2.68·109-s + 3.09·121-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 + 2.15·169-s + 0.0760·173-s + 6.04·175-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 + ⋯

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

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(10.99258117\)
\(L(\frac12)\)  \(\approx\)  \(10.99258117\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.12620587509765358537298960920, −5.72905741876912844479151047101, −5.67649203532205639063722907319, −5.60391034936522234635863257996, −5.29536850351405197216663924773, −5.15271720385637278652934674306, −4.94093571781666379173901085359, −4.69124474677935603357246577137, −4.67943542824686087396002996831, −4.37765547575386927378736418760, −4.27135923135635035578956722115, −3.87974922662895039700121734281, −3.79169159630839195578426690169, −3.32825425642453255345432512781, −3.27906189360233406784391475590, −2.88614031195433916707824735795, −2.79017399054597350358773250514, −2.32824030452573480829642343006, −2.06152133928219905593189017382, −1.85931143963676880871271953232, −1.77288932349726725270360491163, −1.47473377615761563774349049437, −0.930789214913597015010690889754, −0.832079874955314230563478105373, −0.52151947117597763379591725628, 0.52151947117597763379591725628, 0.832079874955314230563478105373, 0.930789214913597015010690889754, 1.47473377615761563774349049437, 1.77288932349726725270360491163, 1.85931143963676880871271953232, 2.06152133928219905593189017382, 2.32824030452573480829642343006, 2.79017399054597350358773250514, 2.88614031195433916707824735795, 3.27906189360233406784391475590, 3.32825425642453255345432512781, 3.79169159630839195578426690169, 3.87974922662895039700121734281, 4.27135923135635035578956722115, 4.37765547575386927378736418760, 4.67943542824686087396002996831, 4.69124474677935603357246577137, 4.94093571781666379173901085359, 5.15271720385637278652934674306, 5.29536850351405197216663924773, 5.60391034936522234635863257996, 5.67649203532205639063722907319, 5.72905741876912844479151047101, 6.12620587509765358537298960920

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