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  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $20.26366152$
$L(\frac12)$  $\approx$  $20.26366152$
$L(\frac{3}{2})$   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

−5.87560489995100177750862077293, −5.27213260918953613100976377152, −5.20022366758776035691150117338, −5.19089157413046600998400348740, −5.11715136489752782023161574461, −4.83408724357781085213142085207, −4.49510082745815113844510194625, −4.31570696992215995681891220499, −4.28791100532694590605530635275, −3.82608333537007061913668484015, −3.73603440455246301396918772860, −3.63214199804410328632377212320, −3.31824731943966133487453945493, −3.08352051768642377856953801326, −2.79699565962237788890786155643, −2.78951034116507858121468846128, −2.52333867345252177263824068008, −1.94372461600315813315499762559, −1.92889648426033721430511828440, −1.75342707279247963894325655693, −1.55068952629075924377673580507, −1.07189988440536400761046579058, −0.911054382644951168779794529568, −0.899808095297293235638379748060, −0.47982309671297790410679821517, 0.47982309671297790410679821517, 0.899808095297293235638379748060, 0.911054382644951168779794529568, 1.07189988440536400761046579058, 1.55068952629075924377673580507, 1.75342707279247963894325655693, 1.92889648426033721430511828440, 1.94372461600315813315499762559, 2.52333867345252177263824068008, 2.78951034116507858121468846128, 2.79699565962237788890786155643, 3.08352051768642377856953801326, 3.31824731943966133487453945493, 3.63214199804410328632377212320, 3.73603440455246301396918772860, 3.82608333537007061913668484015, 4.28791100532694590605530635275, 4.31570696992215995681891220499, 4.49510082745815113844510194625, 4.83408724357781085213142085207, 5.11715136489752782023161574461, 5.19089157413046600998400348740, 5.20022366758776035691150117338, 5.27213260918953613100976377152, 5.87560489995100177750862077293

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