Properties

Label 6-1080e3-1.1-c3e3-0-4
Degree $6$
Conductor $1259712000$
Sign $1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
Motivic weight $3$
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  + 15·5-s + 6·7-s + 12·11-s + 18·13-s − 21·17-s + 57·19-s + 87·23-s + 150·25-s + 138·29-s + 117·31-s + 90·35-s + 150·37-s + 180·43-s + 684·47-s − 537·49-s + 87·53-s + 180·55-s + 714·59-s − 513·61-s + 270·65-s − 174·67-s + 768·71-s − 252·73-s + 72·77-s + 207·79-s + 1.68e3·83-s − 315·85-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.323·7-s + 0.328·11-s + 0.384·13-s − 0.299·17-s + 0.688·19-s + 0.788·23-s + 6/5·25-s + 0.883·29-s + 0.677·31-s + 0.434·35-s + 0.666·37-s + 0.638·43-s + 2.12·47-s − 1.56·49-s + 0.225·53-s + 0.441·55-s + 1.57·59-s − 1.07·61-s + 0.515·65-s − 0.317·67-s + 1.28·71-s − 0.404·73-s + 0.106·77-s + 0.294·79-s + 2.23·83-s − 0.401·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(11.93544232\)
\(L(\frac12)\) \(\approx\) \(11.93544232\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{3} \)
good7$S_4\times C_2$ \( 1 - 6 T + 573 T^{2} - 20 p^{3} T^{3} + 573 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 12 T + 2733 T^{2} - 11024 T^{3} + 2733 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 18 T + 5391 T^{2} - 55708 T^{3} + 5391 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 21 T + 7458 T^{2} + 253565 T^{3} + 7458 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 p T + 8592 T^{2} - 1114405 T^{3} + 8592 p^{3} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 87 T + 28512 T^{2} - 1461623 T^{3} + 28512 p^{3} T^{4} - 87 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 138 T + 20103 T^{2} - 123916 p T^{3} + 20103 p^{3} T^{4} - 138 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 117 T + 20988 T^{2} + 2423207 T^{3} + 20988 p^{3} T^{4} - 117 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 150 T + 95907 T^{2} - 17945796 T^{3} + 95907 p^{3} T^{4} - 150 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 68055 T^{2} + 17332488 T^{3} + 68055 p^{3} T^{4} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 180 T + 22293 T^{2} + 3328688 T^{3} + 22293 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 684 T + 347853 T^{2} - 110757224 T^{3} + 347853 p^{3} T^{4} - 684 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 87 T + 277926 T^{2} + 2139897 T^{3} + 277926 p^{3} T^{4} - 87 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 714 T + 669321 T^{2} - 272637508 T^{3} + 669321 p^{3} T^{4} - 714 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 513 T + 466794 T^{2} + 175503629 T^{3} + 466794 p^{3} T^{4} + 513 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 174 T + 677229 T^{2} + 48960124 T^{3} + 677229 p^{3} T^{4} + 174 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 768 T + 869433 T^{2} - 382859112 T^{3} + 869433 p^{3} T^{4} - 768 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 252 T + 967647 T^{2} + 141584400 T^{3} + 967647 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 207 T + 981660 T^{2} - 306571995 T^{3} + 981660 p^{3} T^{4} - 207 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 1689 T + 2368140 T^{2} - 1880402401 T^{3} + 2368140 p^{3} T^{4} - 1689 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 312 T + 1492215 T^{2} - 319203960 T^{3} + 1492215 p^{3} T^{4} - 312 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1080 T + 1410147 T^{2} + 2105654384 T^{3} + 1410147 p^{3} T^{4} + 1080 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.527622261994652070411136751081, −8.067944450074632518193488584736, −7.912820578663465604694880104542, −7.66062322692297627574796335021, −7.07869709036759986878461173257, −6.94367649972052396858286942535, −6.81228281417323192237787926239, −6.28350199026216973636423694460, −6.09601877424055113368430272342, −5.91368298157705396805582724910, −5.51075954741241688429897535002, −5.14412385176653145781427244210, −5.06202635912384229143057161145, −4.45924689905156759958639343959, −4.36475942072549998046197419671, −4.04632842964869373761866023807, −3.28932787230736124166460666779, −3.20799820932615448082912437565, −2.90730867745021272077376992361, −2.29917349529856224228811743399, −2.02941532605627285752986049316, −1.80643139048891415997666361170, −1.07794441893411459731331292918, −0.789009838882420986469190727256, −0.64426031929350432934824858120, 0.64426031929350432934824858120, 0.789009838882420986469190727256, 1.07794441893411459731331292918, 1.80643139048891415997666361170, 2.02941532605627285752986049316, 2.29917349529856224228811743399, 2.90730867745021272077376992361, 3.20799820932615448082912437565, 3.28932787230736124166460666779, 4.04632842964869373761866023807, 4.36475942072549998046197419671, 4.45924689905156759958639343959, 5.06202635912384229143057161145, 5.14412385176653145781427244210, 5.51075954741241688429897535002, 5.91368298157705396805582724910, 6.09601877424055113368430272342, 6.28350199026216973636423694460, 6.81228281417323192237787926239, 6.94367649972052396858286942535, 7.07869709036759986878461173257, 7.66062322692297627574796335021, 7.912820578663465604694880104542, 8.067944450074632518193488584736, 8.527622261994652070411136751081

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