Properties

Label 6-2160e3-1.1-c3e3-0-12
Degree $6$
Conductor $10077696000$
Sign $-1$
Analytic cond. $2.06994\times 10^{6}$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{9} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(2.06994\times 10^{6}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$S_4\times C_2$ \( 1 + 1080 T + 1410147 T^{2} + 2105654384 T^{3} + 1410147 p^{3} T^{4} + 1080 p^{6} T^{5} + p^{9} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.000830147267542907370752733688, −7.69958205756086549246192114442, −7.69745882535506736497541107243, −7.37301312236406442815372733888, −7.06281835878687276647948262454, −6.74901226599344609439349845533, −6.60935331923612921085350840582, −6.20991084011024603887301856903, −6.16333053197700728864454295621, −5.63172908791594710264928167297, −5.31434426175996028334164554659, −5.17541411240472009708644334159, −4.88910157761808222720075254432, −4.36046533312283327203816935480, −4.27616309266689690879340741374, −3.94006085508132221144180036284, −3.54965535696964745841732183061, −3.44714032179316559707852894206, −3.30091148735065007498839838807, −2.47308569824954038924268770288, −2.43125884001732969422906838093, −2.25642184647317962527735935547, −1.27828366739051079508279276826, −1.17856952881819708577546137022, −1.08476778333450002225523145844, 0, 0, 0, 1.08476778333450002225523145844, 1.17856952881819708577546137022, 1.27828366739051079508279276826, 2.25642184647317962527735935547, 2.43125884001732969422906838093, 2.47308569824954038924268770288, 3.30091148735065007498839838807, 3.44714032179316559707852894206, 3.54965535696964745841732183061, 3.94006085508132221144180036284, 4.27616309266689690879340741374, 4.36046533312283327203816935480, 4.88910157761808222720075254432, 5.17541411240472009708644334159, 5.31434426175996028334164554659, 5.63172908791594710264928167297, 6.16333053197700728864454295621, 6.20991084011024603887301856903, 6.60935331923612921085350840582, 6.74901226599344609439349845533, 7.06281835878687276647948262454, 7.37301312236406442815372733888, 7.69745882535506736497541107243, 7.69958205756086549246192114442, 8.000830147267542907370752733688

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