Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{8} \cdot 7^{8} $
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  + 8·25-s + 16·37-s − 32·43-s − 48·67-s + 16·79-s + 64·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 8/5·25-s + 2.63·37-s − 4.87·43-s − 5.86·67-s + 1.80·79-s + 6.13·109-s + 1.09·121-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 − 1.23·169-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 + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7056} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.6698168057\)
\(L(\frac12)\)  \(\approx\)  \(0.6698168057\)
\(L(\frac{3}{2})\)   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 \( 1 \)
good5$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 164 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.72440668954286310136205193676, −5.33706882625383848980618110027, −5.27483064433548661672910596630, −4.77055487390027247736608970514, −4.75631805664999827923030746277, −4.62879628540677781653199102586, −4.61149936918639295981204792000, −4.37465845947502123117845117777, −4.27422892936224152891349808595, −3.66640374138027896867439052608, −3.59566758787241583009063976354, −3.47120573867133442567222930747, −3.22280337975959876606983494333, −3.03057701853497052232525851509, −2.90083355926649887649209538306, −2.78808889588003421473601151128, −2.42985464080898893740576738446, −1.92585723524760274504635212207, −1.90515901354809279191386696660, −1.77653879951124542232521737707, −1.58999074059699369503867861727, −0.947219629715373193663953557158, −0.940960787235130843704839675487, −0.66777016501471800554745949790, −0.099571781904288462395236872056, 0.099571781904288462395236872056, 0.66777016501471800554745949790, 0.940960787235130843704839675487, 0.947219629715373193663953557158, 1.58999074059699369503867861727, 1.77653879951124542232521737707, 1.90515901354809279191386696660, 1.92585723524760274504635212207, 2.42985464080898893740576738446, 2.78808889588003421473601151128, 2.90083355926649887649209538306, 3.03057701853497052232525851509, 3.22280337975959876606983494333, 3.47120573867133442567222930747, 3.59566758787241583009063976354, 3.66640374138027896867439052608, 4.27422892936224152891349808595, 4.37465845947502123117845117777, 4.61149936918639295981204792000, 4.62879628540677781653199102586, 4.75631805664999827923030746277, 4.77055487390027247736608970514, 5.27483064433548661672910596630, 5.33706882625383848980618110027, 5.72440668954286310136205193676

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