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  − 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

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.631249161\)
\(L(\frac12)\)  \(\approx\)  \(1.631249161\)
\(L(\frac{3}{2})\)   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$C_2^3$ \( 1 + 4 T^{2} - 9 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.24085836357607020114012344346, −5.91693641559906708360944033214, −5.85580522876473977515431457918, −5.77322219928957286700049287249, −5.52446079690529930841310761948, −4.98863158800812318920988707241, −4.91200924686703184812761832225, −4.67573949519844408537769600837, −4.47479126498541948284134646127, −4.36131327197173892145004326711, −4.09404699640716328795613414980, −3.93958572650166429128345566939, −3.73244079010820176763374269038, −3.62489446604934505372337612264, −3.09350204979369570807572078654, −3.05475326010469232405578264598, −2.53810728575792409064889053264, −2.35202285888336777694489587132, −2.14658754433232075816287952598, −2.05333297566573759586525239512, −1.79070178846138437458027219830, −1.59564993816268375051735941066, −0.78213556610527361847047356801, −0.64939869264194115234745103004, −0.25751857565385617252215911255, 0.25751857565385617252215911255, 0.64939869264194115234745103004, 0.78213556610527361847047356801, 1.59564993816268375051735941066, 1.79070178846138437458027219830, 2.05333297566573759586525239512, 2.14658754433232075816287952598, 2.35202285888336777694489587132, 2.53810728575792409064889053264, 3.05475326010469232405578264598, 3.09350204979369570807572078654, 3.62489446604934505372337612264, 3.73244079010820176763374269038, 3.93958572650166429128345566939, 4.09404699640716328795613414980, 4.36131327197173892145004326711, 4.47479126498541948284134646127, 4.67573949519844408537769600837, 4.91200924686703184812761832225, 4.98863158800812318920988707241, 5.52446079690529930841310761948, 5.77322219928957286700049287249, 5.85580522876473977515431457918, 5.91693641559906708360944033214, 6.24085836357607020114012344346

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