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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·25-s + 16·37-s − 24·47-s + 14·49-s − 12·59-s + 48·83-s − 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4/5·25-s + 2.63·37-s − 3.50·47-s + 2·49-s − 1.56·59-s + 5.26·83-s − 6.13·109-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 + 0.769·169-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

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.911456472\)
\(L(\frac12)\)  \(\approx\)  \(2.911456472\)
\(L(\frac{3}{2})\)   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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.98981763710270761208689160428, −5.89345151799220228361544062480, −5.88057113563610627400037320937, −5.87370656747971866759144125921, −5.22563979237517673582218421057, −5.21518871867730154009282005862, −4.81265479800648420960377046923, −4.76335103094274571025763720278, −4.63215516681622433331001492695, −4.38317100606364102124423469455, −4.12535876584937275948195842457, −3.88018829935572687095994075020, −3.54581441697250822623071516613, −3.43075461060917871380850270448, −3.33700483409131087015183657450, −2.99692983859679776130038916266, −2.68168200761265926274422247207, −2.36889557002579166584904714467, −2.21486702787989740790102378284, −2.10826812791739662003575078402, −1.53900610636398936446682359892, −1.44282694542579214612777598933, −1.02634958463653788423674866876, −0.64137665604591634232310830250, −0.30996924983463594574558091552, 0.30996924983463594574558091552, 0.64137665604591634232310830250, 1.02634958463653788423674866876, 1.44282694542579214612777598933, 1.53900610636398936446682359892, 2.10826812791739662003575078402, 2.21486702787989740790102378284, 2.36889557002579166584904714467, 2.68168200761265926274422247207, 2.99692983859679776130038916266, 3.33700483409131087015183657450, 3.43075461060917871380850270448, 3.54581441697250822623071516613, 3.88018829935572687095994075020, 4.12535876584937275948195842457, 4.38317100606364102124423469455, 4.63215516681622433331001492695, 4.76335103094274571025763720278, 4.81265479800648420960377046923, 5.21518871867730154009282005862, 5.22563979237517673582218421057, 5.87370656747971866759144125921, 5.88057113563610627400037320937, 5.89345151799220228361544062480, 5.98981763710270761208689160428

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