Properties

Degree 8
Conductor $ 2^{20} \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  + 2·5-s + 4·7-s + 2·11-s + 8·17-s − 4·19-s + 2·23-s − 4·25-s + 8·29-s + 4·31-s + 8·35-s − 4·37-s + 2·41-s − 8·43-s + 12·47-s + 10·49-s + 16·53-s + 4·55-s + 12·59-s − 8·61-s + 8·67-s − 18·71-s + 8·73-s + 8·77-s + 16·85-s + 18·89-s − 8·95-s + 8·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 0.603·11-s + 1.94·17-s − 0.917·19-s + 0.417·23-s − 4/5·25-s + 1.48·29-s + 0.718·31-s + 1.35·35-s − 0.657·37-s + 0.312·41-s − 1.21·43-s + 1.75·47-s + 10/7·49-s + 2.19·53-s + 0.539·55-s + 1.56·59-s − 1.02·61-s + 0.977·67-s − 2.13·71-s + 0.936·73-s + 0.911·77-s + 1.73·85-s + 1.90·89-s − 0.820·95-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $19.42975516$
$L(\frac12)$  $\approx$  $19.42975516$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} - 24 T^{3} + 33 T^{4} - 24 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 T + 20 T^{2} + 20 T^{3} + 125 T^{4} + 20 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 8 T + 44 T^{2} - 184 T^{3} + 854 T^{4} - 184 p T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 4 T + 34 T^{2} + 72 T^{3} + 699 T^{4} + 72 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} - 4 T^{3} + 761 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 8 T + 80 T^{2} - 296 T^{3} + 2206 T^{4} - 296 p T^{5} + 80 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4 T + 70 T^{2} - 448 T^{3} + 2407 T^{4} - 448 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 344 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2 T + 92 T^{2} - 80 T^{3} + 4093 T^{4} - 80 p T^{5} + 92 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 8 T + 160 T^{2} + 952 T^{3} + 10046 T^{4} + 952 p T^{5} + 160 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 12 T + 200 T^{2} - 1484 T^{3} + 13950 T^{4} - 1484 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
59$C_2 \wr S_4$ \( 1 - 12 T + 248 T^{2} - 1916 T^{3} + 21870 T^{4} - 1916 p T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 440 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 8 T + 88 T^{2} - 8 T^{3} + 814 T^{4} - 8 p T^{5} + 88 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 18 T + 296 T^{2} + 2724 T^{3} + 27369 T^{4} + 2724 p T^{5} + 296 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 1032 p T^{5} + 136 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 160 T^{2} + 848 T^{3} + 11886 T^{4} + 848 p T^{5} + 160 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 40 T^{2} - 16 T^{3} + 13134 T^{4} - 16 p T^{5} - 40 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 18 T + 404 T^{2} - 4416 T^{3} + 54549 T^{4} - 4416 p T^{5} + 404 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 2200 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.81986758480460425902272920511, −5.48178597320630091026664627929, −5.33663642764685398633458449749, −5.09154026916829346181855009788, −5.04258058462312142321834636916, −4.58429676106604564289113729353, −4.52029539024521554909237919194, −4.51654049056151428209862534762, −4.27818018733634369590465635056, −3.84150496388800565339534271506, −3.65079969531765684506036754988, −3.62541178626573742165541987047, −3.53468406791929501137330882824, −2.85423037573507499170664593970, −2.82455982816561083249617642606, −2.79514425384011553536546920670, −2.51391362325707273568264638449, −1.84421145617305524027657461211, −1.84225893624531802443640641417, −1.83896078671121413008254932303, −1.81830509337578447726443994116, −1.09260688759613037508718274949, −0.818436406134259403692567725934, −0.75178492193240672237652132633, −0.56381333288800414125007145716, 0.56381333288800414125007145716, 0.75178492193240672237652132633, 0.818436406134259403692567725934, 1.09260688759613037508718274949, 1.81830509337578447726443994116, 1.83896078671121413008254932303, 1.84225893624531802443640641417, 1.84421145617305524027657461211, 2.51391362325707273568264638449, 2.79514425384011553536546920670, 2.82455982816561083249617642606, 2.85423037573507499170664593970, 3.53468406791929501137330882824, 3.62541178626573742165541987047, 3.65079969531765684506036754988, 3.84150496388800565339534271506, 4.27818018733634369590465635056, 4.51654049056151428209862534762, 4.52029539024521554909237919194, 4.58429676106604564289113729353, 5.04258058462312142321834636916, 5.09154026916829346181855009788, 5.33663642764685398633458449749, 5.48178597320630091026664627929, 5.81986758480460425902272920511

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