Properties

Label 6-50e3-1.1-c21e3-0-0
Degree $6$
Conductor $125000$
Sign $-1$
Analytic cond. $2.72866\times 10^{6}$
Root an. cond. $11.8211$
Motivic weight $21$
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  − 3.07e3·2-s − 4.63e4·3-s + 6.29e6·4-s + 1.42e8·6-s + 9.11e8·7-s − 1.07e10·8-s − 2.08e9·9-s − 7.75e10·11-s − 2.91e11·12-s − 2.93e10·13-s − 2.80e12·14-s + 1.64e13·16-s + 1.88e13·17-s + 6.40e12·18-s − 2.39e13·19-s − 4.22e13·21-s + 2.38e14·22-s − 2.02e14·23-s + 4.98e14·24-s + 9.00e13·26-s + 5.49e13·27-s + 5.73e15·28-s + 3.18e15·29-s − 2.95e15·31-s − 2.36e16·32-s + 3.59e15·33-s − 5.77e16·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.453·3-s + 3·4-s + 0.962·6-s + 1.21·7-s − 3.53·8-s − 0.199·9-s − 0.901·11-s − 1.36·12-s − 0.0589·13-s − 2.58·14-s + 15/4·16-s + 2.26·17-s + 0.422·18-s − 0.896·19-s − 0.553·21-s + 1.91·22-s − 1.01·23-s + 1.60·24-s + 0.125·26-s + 0.0513·27-s + 3.65·28-s + 1.40·29-s − 0.646·31-s − 3.71·32-s + 0.408·33-s − 4.80·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125000 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(125000\)    =    \(2^{3} \cdot 5^{6}\)
Sign: $-1$
Analytic conductor: \(2.72866\times 10^{6}\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 125000,\ (\ :21/2, 21/2, 21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 + p^{10} T )^{3} \)
5 \( 1 \)
good3$S_4\times C_2$ \( 1 + 15461 p T + 52281512 p^{4} T^{2} + 36293334139 p^{8} T^{3} + 52281512 p^{25} T^{4} + 15461 p^{43} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 911775234 T + 24563517932509377 p^{2} T^{2} - \)\(21\!\cdots\!72\)\( p^{2} T^{3} + 24563517932509377 p^{23} T^{4} - 911775234 p^{42} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 + 77565926349 T + \)\(20\!\cdots\!00\)\( p T^{2} + \)\(94\!\cdots\!65\)\( p^{2} T^{3} + \)\(20\!\cdots\!00\)\( p^{22} T^{4} + 77565926349 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 + 29305708548 T + \)\(60\!\cdots\!07\)\( T^{2} + \)\(21\!\cdots\!88\)\( p T^{3} + \)\(60\!\cdots\!07\)\( p^{21} T^{4} + 29305708548 p^{42} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 - 1106562440457 p T + \)\(57\!\cdots\!26\)\( p^{3} T^{2} - \)\(50\!\cdots\!01\)\( p^{3} T^{3} + \)\(57\!\cdots\!26\)\( p^{24} T^{4} - 1106562440457 p^{43} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 + 1261032950775 p T + \)\(48\!\cdots\!12\)\( p^{2} T^{2} + \)\(36\!\cdots\!75\)\( p^{3} T^{3} + \)\(48\!\cdots\!12\)\( p^{23} T^{4} + 1261032950775 p^{43} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 + 202301945446698 T + \)\(74\!\cdots\!37\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(74\!\cdots\!37\)\( p^{21} T^{4} + 202301945446698 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 - 3183619096555920 T + \)\(83\!\cdots\!87\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!87\)\( p^{21} T^{4} - 3183619096555920 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 + 2950614234339474 T + \)\(23\!\cdots\!85\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!85\)\( p^{21} T^{4} + 2950614234339474 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 - 29769971443997754 T + \)\(27\!\cdots\!83\)\( T^{2} - \)\(51\!\cdots\!28\)\( T^{3} + \)\(27\!\cdots\!83\)\( p^{21} T^{4} - 29769971443997754 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 + 189362455401586329 T + \)\(28\!\cdots\!70\)\( T^{2} + \)\(25\!\cdots\!85\)\( T^{3} + \)\(28\!\cdots\!70\)\( p^{21} T^{4} + 189362455401586329 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 + 186191720065045428 T + \)\(67\!\cdots\!57\)\( T^{2} + \)\(74\!\cdots\!84\)\( T^{3} + \)\(67\!\cdots\!57\)\( p^{21} T^{4} + 186191720065045428 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 - 154627401898247004 T + \)\(33\!\cdots\!13\)\( T^{2} + \)\(60\!\cdots\!92\)\( T^{3} + \)\(33\!\cdots\!13\)\( p^{21} T^{4} - 154627401898247004 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 - 636975366021883422 T + \)\(41\!\cdots\!87\)\( T^{2} - \)\(21\!\cdots\!56\)\( T^{3} + \)\(41\!\cdots\!87\)\( p^{21} T^{4} - 636975366021883422 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 + 8039229184581194160 T + \)\(37\!\cdots\!77\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!77\)\( p^{21} T^{4} + 8039229184581194160 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 - 2623888488791543046 T + \)\(37\!\cdots\!55\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!55\)\( p^{21} T^{4} - 2623888488791543046 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 - 14413057977134390259 T + \)\(42\!\cdots\!28\)\( T^{2} - \)\(47\!\cdots\!83\)\( T^{3} + \)\(42\!\cdots\!28\)\( p^{21} T^{4} - 14413057977134390259 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 50615572234903950276 T + \)\(28\!\cdots\!05\)\( T^{2} - \)\(77\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!05\)\( p^{21} T^{4} - 50615572234903950276 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 - 34508521524225003807 T + \)\(20\!\cdots\!02\)\( T^{2} - \)\(20\!\cdots\!31\)\( T^{3} + \)\(20\!\cdots\!02\)\( p^{21} T^{4} - 34508521524225003807 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(23\!\cdots\!70\)\( T + \)\(39\!\cdots\!37\)\( T^{2} + \)\(38\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!37\)\( p^{21} T^{4} + \)\(23\!\cdots\!70\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 - 69706807117815066327 T + \)\(49\!\cdots\!92\)\( T^{2} - \)\(30\!\cdots\!11\)\( T^{3} + \)\(49\!\cdots\!92\)\( p^{21} T^{4} - 69706807117815066327 p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(64\!\cdots\!55\)\( T + \)\(35\!\cdots\!42\)\( T^{2} + \)\(11\!\cdots\!15\)\( T^{3} + \)\(35\!\cdots\!42\)\( p^{21} T^{4} + \)\(64\!\cdots\!55\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(57\!\cdots\!94\)\( T + \)\(82\!\cdots\!03\)\( T^{2} - \)\(59\!\cdots\!28\)\( T^{3} + \)\(82\!\cdots\!03\)\( p^{21} T^{4} - \)\(57\!\cdots\!94\)\( p^{42} T^{5} + p^{63} T^{6} \)
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

−10.26975999712600125515900031242, −10.14445083222343657189088324465, −9.690597603707867753355766770219, −9.598098530538008779849050260262, −8.702165787884329190164874207352, −8.405348027340301702931076139309, −8.346769066769793840433280836424, −7.941534910388565926077958438037, −7.48474831013476155010302236992, −7.44256496919471901821185147194, −6.66515051395736423201916314307, −6.31197268320744078391074406137, −6.09484323027338010578384412808, −5.29128998723217115911575277606, −5.28445522119240856399293614839, −4.87277969914629137265234236255, −4.14880877057248408840427733886, −3.53978927045032563411293712920, −3.26948256509761780771398708404, −2.57060912598023625161112233154, −2.44595973165407049278257057506, −1.85542861813644868843119807596, −1.47661445193045459318255441358, −1.10964839751404106077221278193, −1.01763923328993141198890654935, 0, 0, 0, 1.01763923328993141198890654935, 1.10964839751404106077221278193, 1.47661445193045459318255441358, 1.85542861813644868843119807596, 2.44595973165407049278257057506, 2.57060912598023625161112233154, 3.26948256509761780771398708404, 3.53978927045032563411293712920, 4.14880877057248408840427733886, 4.87277969914629137265234236255, 5.28445522119240856399293614839, 5.29128998723217115911575277606, 6.09484323027338010578384412808, 6.31197268320744078391074406137, 6.66515051395736423201916314307, 7.44256496919471901821185147194, 7.48474831013476155010302236992, 7.941534910388565926077958438037, 8.346769066769793840433280836424, 8.405348027340301702931076139309, 8.702165787884329190164874207352, 9.598098530538008779849050260262, 9.690597603707867753355766770219, 10.14445083222343657189088324465, 10.26975999712600125515900031242

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