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  + 4·7-s − 14·25-s + 4·37-s − 28·43-s − 2·49-s + 40·67-s − 20·79-s + 52·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s − 56·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.51·7-s − 2.79·25-s + 0.657·37-s − 4.26·43-s − 2/7·49-s + 4.88·67-s − 2.25·79-s + 4.98·109-s + 8/11·121-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 − 4.23·175-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  =  \((8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(4.554318978\)
\(L(\frac12)\)  \(\approx\)  \(4.554318978\)
\(L(\frac{3}{2})\)   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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 188 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.21208742997116623237146350222, −5.87058944076916705566913397397, −5.70893574452764123888537827355, −5.63393415860003010802492175616, −5.33164673248896392677464896767, −5.21012889090947761970081452266, −4.86566877578844198242221059250, −4.77448678283064511973801215807, −4.58848325592802861323520408796, −4.39680392659725505441431372008, −4.15015694087300773238637750350, −3.89797676505784121590230570039, −3.57869965554652195770290364262, −3.47333657536795122779936744238, −3.32432077125894175018974792028, −2.89279373750426527116407326326, −2.88737819931145486457547640673, −2.21938350226781555797689071923, −2.01208597523022734434091820916, −1.95610039643426885534331612010, −1.75770427776262166650486565344, −1.59709802105164168344297905258, −1.03221639396362627318288334790, −0.55503927228159515059183686625, −0.41978416099709583996406668142, 0.41978416099709583996406668142, 0.55503927228159515059183686625, 1.03221639396362627318288334790, 1.59709802105164168344297905258, 1.75770427776262166650486565344, 1.95610039643426885534331612010, 2.01208597523022734434091820916, 2.21938350226781555797689071923, 2.88737819931145486457547640673, 2.89279373750426527116407326326, 3.32432077125894175018974792028, 3.47333657536795122779936744238, 3.57869965554652195770290364262, 3.89797676505784121590230570039, 4.15015694087300773238637750350, 4.39680392659725505441431372008, 4.58848325592802861323520408796, 4.77448678283064511973801215807, 4.86566877578844198242221059250, 5.21012889090947761970081452266, 5.33164673248896392677464896767, 5.63393415860003010802492175616, 5.70893574452764123888537827355, 5.87058944076916705566913397397, 6.21208742997116623237146350222

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