Properties

Label 6-8280e3-1.1-c1e3-0-3
Degree $6$
Conductor $567663552000$
Sign $1$
Analytic cond. $289016.$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 4·7-s + 6·11-s + 4·17-s + 4·19-s − 3·23-s + 6·25-s − 4·29-s − 2·31-s + 12·35-s + 2·37-s + 14·41-s + 16·47-s − 4·49-s + 4·53-s + 18·55-s + 4·59-s + 14·61-s + 6·67-s + 10·71-s + 10·73-s + 24·77-s − 4·79-s + 10·83-s + 12·85-s − 20·89-s + 12·95-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.51·7-s + 1.80·11-s + 0.970·17-s + 0.917·19-s − 0.625·23-s + 6/5·25-s − 0.742·29-s − 0.359·31-s + 2.02·35-s + 0.328·37-s + 2.18·41-s + 2.33·47-s − 4/7·49-s + 0.549·53-s + 2.42·55-s + 0.520·59-s + 1.79·61-s + 0.733·67-s + 1.18·71-s + 1.17·73-s + 2.73·77-s − 0.450·79-s + 1.09·83-s + 1.30·85-s − 2.11·89-s + 1.23·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(289016.\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.98065944\)
\(L(\frac12)\) \(\approx\) \(16.98065944\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - 4 T + 20 T^{2} - 54 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 6 T + 35 T^{2} - 112 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 29 T^{2} + 8 T^{3} + 29 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 50 T^{2} - 134 T^{3} + 50 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 4 T + 43 T^{2} - 160 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 84 T^{2} + 222 T^{3} + 84 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 86 T^{2} + 120 T^{3} + 86 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 106 T^{2} - 140 T^{3} + 106 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 180 T^{2} - 1212 T^{3} + 180 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 89 T^{2} + 64 T^{3} + 89 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 16 T + 207 T^{2} - 1584 T^{3} + 207 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 4 T + 158 T^{2} - 422 T^{3} + 158 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 110 T^{2} - 614 T^{3} + 110 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 181 T^{2} - 1640 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T + 92 T^{2} - 824 T^{3} + 92 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 10 T + 14 T^{2} + 448 T^{3} + 14 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 89 T^{2} - 696 T^{3} + 89 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 4 T + 109 T^{2} + 376 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 252 T^{2} - 1656 T^{3} + 252 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 20 T + 323 T^{2} + 3592 T^{3} + 323 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 10 T + 247 T^{2} + 1468 T^{3} + 247 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.99987345705951829784033869871, −6.58184267633886645665168857692, −6.34699984018479211499379168701, −6.21551943475401613350253780204, −5.95363595535186813464035634779, −5.72227842405176980325646295639, −5.48251248725833463906869734343, −5.22872147407285261780014487211, −5.07906816794478920043471446268, −5.03795435440640961664513496202, −4.32447679290762207484901149499, −4.24875028426973949862113138864, −4.18636207563779113946576057425, −3.72196090887798861878111956788, −3.48223805731530195791643407367, −3.46596337953633871809561429491, −2.71559181380713158830666389591, −2.50158112456511000198698090937, −2.49223560711058444025871478612, −1.94074602795656891406402317046, −1.68497910534793668716666021733, −1.56388609706987020206002817225, −1.08771509451387408035823179124, −0.814168308054472392000466319306, −0.66122410773765381360007950129, 0.66122410773765381360007950129, 0.814168308054472392000466319306, 1.08771509451387408035823179124, 1.56388609706987020206002817225, 1.68497910534793668716666021733, 1.94074602795656891406402317046, 2.49223560711058444025871478612, 2.50158112456511000198698090937, 2.71559181380713158830666389591, 3.46596337953633871809561429491, 3.48223805731530195791643407367, 3.72196090887798861878111956788, 4.18636207563779113946576057425, 4.24875028426973949862113138864, 4.32447679290762207484901149499, 5.03795435440640961664513496202, 5.07906816794478920043471446268, 5.22872147407285261780014487211, 5.48251248725833463906869734343, 5.72227842405176980325646295639, 5.95363595535186813464035634779, 6.21551943475401613350253780204, 6.34699984018479211499379168701, 6.58184267633886645665168857692, 6.99987345705951829784033869871

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