Properties

Label 6-2760e3-1.1-c1e3-0-1
Degree $6$
Conductor $21024576000$
Sign $1$
Analytic cond. $10704.3$
Root an. cond. $4.69454$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·5-s + 7-s + 6·9-s + 4·13-s + 9·15-s + 7·17-s + 3·21-s + 3·23-s + 6·25-s + 10·27-s + 5·29-s + 31-s + 3·35-s + 9·37-s + 12·39-s + 3·41-s + 18·45-s + 2·47-s − 8·49-s + 21·51-s + 15·53-s + 3·59-s − 6·61-s + 6·63-s + 12·65-s − 3·67-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 1.10·13-s + 2.32·15-s + 1.69·17-s + 0.654·21-s + 0.625·23-s + 6/5·25-s + 1.92·27-s + 0.928·29-s + 0.179·31-s + 0.507·35-s + 1.47·37-s + 1.92·39-s + 0.468·41-s + 2.68·45-s + 0.291·47-s − 8/7·49-s + 2.94·51-s + 2.06·53-s + 0.390·59-s − 0.768·61-s + 0.755·63-s + 1.48·65-s − 0.366·67-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(10704.3\)
Root analytic conductor: \(4.69454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.00259987\)
\(L(\frac12)\) \(\approx\) \(19.00259987\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 - T + 9 T^{2} + 2 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
13$D_{6}$ \( 1 - 4 T + 19 T^{2} - 88 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 7 T + 55 T^{2} - 210 T^{3} + 55 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
19$C_2$ \( ( 1 + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 - 5 T + 83 T^{2} - 286 T^{3} + 83 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - T + 61 T^{2} + 2 T^{3} + 61 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 9 T + 35 T^{2} + 10 T^{3} + 35 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 3 T + 83 T^{2} - 98 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 93 T^{2} - 60 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 15 T + 191 T^{2} - 1394 T^{3} + 191 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 3 T + 137 T^{2} - 418 T^{3} + 137 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 131 T^{2} + 548 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 3 T + 101 T^{2} - 94 T^{3} + 101 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 5 T + 97 T^{2} - 822 T^{3} + 97 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 8 T + 191 T^{2} - 960 T^{3} + 191 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 8 T + 173 T^{2} - 816 T^{3} + 173 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 141 T^{2} - 570 T^{3} + 141 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 20 T + 271 T^{2} - 2568 T^{3} + 271 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 239 T^{2} - 980 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.962740691291711267614675053910, −7.57371601122615747973926052969, −7.41543501713353955659721091576, −7.26468172921653688586343477190, −6.63028149616968766806720444926, −6.51564650367522708173332042238, −6.43183885832394844288862905047, −6.02403188654542751428581729134, −5.73743678102741873679482828906, −5.54598254167902757241732005586, −4.97067616329237072165063894044, −4.96237404235056616861471969400, −4.81523533494161858235634675448, −4.10022832181401538336552718079, −3.93034566409509717222496427382, −3.83022006134738278016562742401, −3.25811507196515210993208745274, −2.99314347025409136142181810938, −2.97407601419974883919205807925, −2.37044060451825392700643278334, −2.12290330002362793953796179573, −1.98045531156101823126649434048, −1.17840817945433753252223274223, −1.13662971583271083591384289217, −0.883436019144187733132308289500, 0.883436019144187733132308289500, 1.13662971583271083591384289217, 1.17840817945433753252223274223, 1.98045531156101823126649434048, 2.12290330002362793953796179573, 2.37044060451825392700643278334, 2.97407601419974883919205807925, 2.99314347025409136142181810938, 3.25811507196515210993208745274, 3.83022006134738278016562742401, 3.93034566409509717222496427382, 4.10022832181401538336552718079, 4.81523533494161858235634675448, 4.96237404235056616861471969400, 4.97067616329237072165063894044, 5.54598254167902757241732005586, 5.73743678102741873679482828906, 6.02403188654542751428581729134, 6.43183885832394844288862905047, 6.51564650367522708173332042238, 6.63028149616968766806720444926, 7.26468172921653688586343477190, 7.41543501713353955659721091576, 7.57371601122615747973926052969, 7.962740691291711267614675053910

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