Properties

Degree 8
Conductor $ 2^{8} \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  − 2·2-s − 4-s + 8·8-s − 7·16-s + 14·25-s − 20·29-s − 14·32-s − 28·50-s − 28·53-s + 40·58-s + 35·64-s − 14·100-s + 56·106-s + 56·113-s + 20·116-s + 30·121-s + 127-s − 4·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s − 7/4·16-s + 14/5·25-s − 3.71·29-s − 2.47·32-s − 3.95·50-s − 3.84·53-s + 5.25·58-s + 35/8·64-s − 7/5·100-s + 5.43·106-s + 5.26·113-s + 1.85·116-s + 2.72·121-s + 0.0887·127-s − 0.353·128-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 + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.7295168279\)
\(L(\frac12)\)  \(\approx\)  \(0.7295168279\)
\(L(\frac{3}{2})\)   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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 15 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 145 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.58356058048873490179702537438, −6.49331606650418778240929965600, −6.22847956775777767490732270768, −5.99289540515879073945625565177, −5.84240539064496304283735799790, −5.43638322239183032614639378022, −5.20182501429457144821064245584, −5.09983209461365445667346049628, −5.06080153726314526007437265660, −4.50506577477380432737632019772, −4.41061637432354085562968578428, −4.38328095269047818774587909125, −4.14109283105134768006266820139, −3.46455425655329891507912550135, −3.35220620176171747092855672749, −3.32524312924954652001708036831, −3.30426640783988546062337694758, −2.54510329369379419553273193999, −2.21748186879031739658147967852, −2.05515732301239652463574988133, −1.58464797345731596690516060120, −1.46131773783984180374372413399, −1.14062674875724875949034365782, −0.50966949400204284487683488126, −0.36522061699548891888520923289, 0.36522061699548891888520923289, 0.50966949400204284487683488126, 1.14062674875724875949034365782, 1.46131773783984180374372413399, 1.58464797345731596690516060120, 2.05515732301239652463574988133, 2.21748186879031739658147967852, 2.54510329369379419553273193999, 3.30426640783988546062337694758, 3.32524312924954652001708036831, 3.35220620176171747092855672749, 3.46455425655329891507912550135, 4.14109283105134768006266820139, 4.38328095269047818774587909125, 4.41061637432354085562968578428, 4.50506577477380432737632019772, 5.06080153726314526007437265660, 5.09983209461365445667346049628, 5.20182501429457144821064245584, 5.43638322239183032614639378022, 5.84240539064496304283735799790, 5.99289540515879073945625565177, 6.22847956775777767490732270768, 6.49331606650418778240929965600, 6.58356058048873490179702537438

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