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 − 4·13-s + 4·17-s − 4·19-s + 10·23-s − 4·25-s − 4·29-s + 8·31-s + 8·35-s − 8·37-s + 2·41-s + 4·43-s + 8·47-s + 10·49-s + 16·53-s + 4·55-s + 24·59-s − 8·61-s − 8·65-s − 4·67-s + 10·71-s + 4·73-s + 8·77-s − 4·79-s + 20·83-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 0.603·11-s − 1.10·13-s + 0.970·17-s − 0.917·19-s + 2.08·23-s − 4/5·25-s − 0.742·29-s + 1.43·31-s + 1.35·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 10/7·49-s + 2.19·53-s + 0.539·55-s + 3.12·59-s − 1.02·61-s − 0.992·65-s − 0.488·67-s + 1.18·71-s + 0.468·73-s + 0.911·77-s − 0.450·79-s + 2.19·83-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  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $13.58696157$
$L(\frac12)$  $\approx$  $13.58696157$
$L(\frac{3}{2})$   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$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} - 4 p T^{3} + 57 T^{4} - 4 p^{2} T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 T + 12 T^{2} + 48 T^{3} - 51 T^{4} + 48 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} - 20 T^{3} - 58 T^{4} - 20 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 36 T^{2} - 124 T^{3} + 694 T^{4} - 124 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 4 T + 26 T^{2} - 40 T^{3} + 3 T^{4} - 40 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 10 T + 96 T^{2} - 576 T^{3} + 3385 T^{4} - 576 p T^{5} + 96 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 76 T^{2} + 172 T^{3} + 2694 T^{4} + 172 p T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 8 T + 94 T^{2} - 392 T^{3} + 3303 T^{4} - 392 p T^{5} + 94 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 8 T + 62 T^{2} - 40 T^{3} - 121 T^{4} - 40 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2 T + 108 T^{2} - 4 T^{3} + 5237 T^{4} - 4 p T^{5} + 108 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 4 T + 68 T^{2} + 220 T^{3} + 1014 T^{4} + 220 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 8 T + 124 T^{2} - 1016 T^{3} + 7926 T^{4} - 1016 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 148 T^{2} - 1136 T^{3} + 8534 T^{4} - 1136 p T^{5} + 148 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 24 T + 348 T^{2} - 3464 T^{3} + 29158 T^{4} - 3464 p T^{5} + 348 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 116 T^{2} + 824 T^{3} + 7478 T^{4} + 824 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 4 T + 100 T^{2} + 692 T^{3} + 4454 T^{4} + 692 p T^{5} + 100 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 10 T + 192 T^{2} - 1424 T^{3} + 19193 T^{4} - 1424 p T^{5} + 192 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 4 T + 44 T^{2} - 268 T^{3} + 10310 T^{4} - 268 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 4 T + 20 T^{2} + 404 T^{3} + 11814 T^{4} + 404 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 20 T + 436 T^{2} - 4932 T^{3} + 57734 T^{4} - 4932 p T^{5} + 436 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 26 T + 548 T^{2} - 7140 T^{3} + 80541 T^{4} - 7140 p T^{5} + 548 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 8 T + 196 T^{2} - 1688 T^{3} + 26118 T^{4} - 1688 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.65135577546532276114169803048, −5.28464566558142230004238750780, −5.26956982785922822072754729908, −5.23103120337123615718147644087, −5.08099565591386451220694979339, −4.77420582559853547978653490445, −4.46194093593654036327415215955, −4.39972410432937761696398045499, −4.32539563792993796321515971365, −3.74497515801079337717578736923, −3.73256158071507498066341128953, −3.70060737761899590318465919726, −3.49323672966653487714372092501, −3.01306509434009332523956426195, −2.67809700903421821754029440984, −2.56575562417848725922554184361, −2.56166471135152656566049601942, −2.15119820778900317536494058085, −1.93389163712046863139734171199, −1.74481064213271235360648328564, −1.67594470274124106960422845604, −1.00133583965082406942852965322, −0.973542702244183200798730463425, −0.74367617405895369296108230771, −0.43285004954102353051739162450, 0.43285004954102353051739162450, 0.74367617405895369296108230771, 0.973542702244183200798730463425, 1.00133583965082406942852965322, 1.67594470274124106960422845604, 1.74481064213271235360648328564, 1.93389163712046863139734171199, 2.15119820778900317536494058085, 2.56166471135152656566049601942, 2.56575562417848725922554184361, 2.67809700903421821754029440984, 3.01306509434009332523956426195, 3.49323672966653487714372092501, 3.70060737761899590318465919726, 3.73256158071507498066341128953, 3.74497515801079337717578736923, 4.32539563792993796321515971365, 4.39972410432937761696398045499, 4.46194093593654036327415215955, 4.77420582559853547978653490445, 5.08099565591386451220694979339, 5.23103120337123615718147644087, 5.26956982785922822072754729908, 5.28464566558142230004238750780, 5.65135577546532276114169803048

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