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 + 8·13-s − 2·17-s − 8·19-s − 25-s − 6·29-s − 6·31-s + 8·35-s + 6·37-s + 6·41-s + 2·43-s − 2·47-s + 10·49-s − 6·53-s + 4·61-s − 16·65-s + 4·67-s − 16·71-s + 12·73-s + 2·79-s + 18·83-s + 4·85-s + 8·89-s − 32·91-s + 16·95-s + 24·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 2.21·13-s − 0.485·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.07·31-s + 1.35·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s − 0.291·47-s + 10/7·49-s − 0.824·53-s + 0.512·61-s − 1.98·65-s + 0.488·67-s − 1.89·71-s + 1.40·73-s + 0.225·79-s + 1.97·83-s + 0.433·85-s + 0.847·89-s − 3.35·91-s + 1.64·95-s + 2.43·97-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  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.842151047$
$L(\frac12)$  $\approx$  $1.842151047$
$L(\frac{3}{2})$   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$C_2 \wr S_4$ \( 1 + 2 T + p T^{2} + 18 T^{3} + 48 T^{4} + 18 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 8 T^{2} + 16 T^{3} + 126 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 8 T + 31 T^{2} - 128 T^{3} + 556 T^{4} - 128 p T^{5} + 31 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 2 T + 20 T^{2} - 52 T^{3} + 13 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 8 T + 64 T^{2} + 360 T^{3} + 1806 T^{4} + 360 p T^{5} + 64 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 47 T^{2} - 128 T^{3} + 1008 T^{4} - 128 p T^{5} + 47 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 6 T + 113 T^{2} + 490 T^{3} + 4896 T^{4} + 490 p T^{5} + 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 6 T + 121 T^{2} + 526 T^{3} + 5604 T^{4} + 526 p T^{5} + 121 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 6 T + 97 T^{2} - 262 T^{3} + 3804 T^{4} - 262 p T^{5} + 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 6 T + 161 T^{2} - 698 T^{3} + 9852 T^{4} - 698 p T^{5} + 161 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 2 T + 100 T^{2} + 104 T^{3} + 4469 T^{4} + 104 p T^{5} + 100 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 2 T + 41 T^{2} + 146 T^{3} + 3640 T^{4} + 146 p T^{5} + 41 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 6 T + p T^{2} - 370 T^{3} - 2412 T^{4} - 370 p T^{5} + p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 170 T^{2} + 32 T^{3} + 13899 T^{4} + 32 p T^{5} + 170 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} + 116 T^{3} + 6358 T^{4} + 116 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 4 T + 139 T^{2} - 892 T^{3} + 10912 T^{4} - 892 p T^{5} + 139 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 16 T + 335 T^{2} + 3432 T^{3} + 37440 T^{4} + 3432 p T^{5} + 335 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 12 T + 184 T^{2} - 2148 T^{3} + 16782 T^{4} - 2148 p T^{5} + 184 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$S_4\times C_2$ \( 1 - 2 T + 97 T^{2} + 1498 T^{3} - 1952 T^{4} + 1498 p T^{5} + 97 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 18 T + 389 T^{2} - 4490 T^{3} + 50748 T^{4} - 4490 p T^{5} + 389 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 8 T + 263 T^{2} - 1500 T^{3} + 30168 T^{4} - 1500 p T^{5} + 263 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 24 T + 460 T^{2} - 5704 T^{3} + 65814 T^{4} - 5704 p T^{5} + 460 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.87976555491823846728735587947, −5.45772182846955295634266443687, −5.34178357917048277128398504645, −5.11576257541544519760119368618, −5.05617150893698857875816532821, −4.50532685950744910243341219794, −4.33812409878164773533152275671, −4.28922706954406643639333691619, −4.25080306713208167534414791718, −3.77246648103971959206995557915, −3.67400830954429388307782575039, −3.66607813458766856062279296097, −3.55907727146283424361916910823, −3.04987047852037760522427943738, −2.92396102183980194736626359892, −2.78221419890145217350491935060, −2.62541738646179927831798451508, −1.99003652134668059131515679829, −1.87128340978240931319335659275, −1.82059380546967648439422623439, −1.75423328404891786128253559311, −0.922085834954030413294704919068, −0.802543593110700039256812044837, −0.57199795422711043454293986921, −0.24363135912001096553816149777, 0.24363135912001096553816149777, 0.57199795422711043454293986921, 0.802543593110700039256812044837, 0.922085834954030413294704919068, 1.75423328404891786128253559311, 1.82059380546967648439422623439, 1.87128340978240931319335659275, 1.99003652134668059131515679829, 2.62541738646179927831798451508, 2.78221419890145217350491935060, 2.92396102183980194736626359892, 3.04987047852037760522427943738, 3.55907727146283424361916910823, 3.66607813458766856062279296097, 3.67400830954429388307782575039, 3.77246648103971959206995557915, 4.25080306713208167534414791718, 4.28922706954406643639333691619, 4.33812409878164773533152275671, 4.50532685950744910243341219794, 5.05617150893698857875816532821, 5.11576257541544519760119368618, 5.34178357917048277128398504645, 5.45772182846955295634266443687, 5.87976555491823846728735587947

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