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  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $8.196627145$
$L(\frac12)$  $\approx$  $8.196627145$
$L(\frac{3}{2})$   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

−5.80057510598067748697111354935, −5.54149661834223265350378098513, −5.37660692987237350587053113195, −5.27026939780687206005627221816, −4.96793492206244362741987392000, −4.71050144729925988210262325140, −4.34437391611876774522119218810, −4.34033924726197310320849410639, −4.20239713594692921139778733437, −3.77152901042249373005512248827, −3.67528893612957967731653905521, −3.64602330079433473095398389231, −3.46193659640465037868618973889, −3.03050387615154935024731562973, −2.89152206441950383685451647310, −2.67186673837558427238169262753, −2.58595649955348416239391220494, −2.12507417857292595973522111197, −1.87621594815001245360207698438, −1.76735063092266528693842950308, −1.67030226839695785352455612645, −1.15425268866883947242305371601, −0.67621378388326198620621996348, −0.63630450033341741494164250736, −0.47280832084283859974729128156, 0.47280832084283859974729128156, 0.63630450033341741494164250736, 0.67621378388326198620621996348, 1.15425268866883947242305371601, 1.67030226839695785352455612645, 1.76735063092266528693842950308, 1.87621594815001245360207698438, 2.12507417857292595973522111197, 2.58595649955348416239391220494, 2.67186673837558427238169262753, 2.89152206441950383685451647310, 3.03050387615154935024731562973, 3.46193659640465037868618973889, 3.64602330079433473095398389231, 3.67528893612957967731653905521, 3.77152901042249373005512248827, 4.20239713594692921139778733437, 4.34033924726197310320849410639, 4.34437391611876774522119218810, 4.71050144729925988210262325140, 4.96793492206244362741987392000, 5.27026939780687206005627221816, 5.37660692987237350587053113195, 5.54149661834223265350378098513, 5.80057510598067748697111354935

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