↑

Properties

Label 6-2e9-1.1-c23e3-0-1
Degree $6$
Conductor $512$
Sign $-1$
Analytic cond. $19284.0$
Root an. cond. $5.17844$
Motivic weight $23$
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  − 2.13e5·3-s + 9.56e7·5-s − 8.64e9·7-s − 3.10e10·9-s + 3.54e10·11-s + 3.16e12·13-s − 2.04e13·15-s − 3.02e13·17-s − 3.82e14·19-s + 1.85e15·21-s − 3.75e15·23-s − 2.20e16·25-s − 1.79e16·27-s − 1.42e17·29-s − 2.04e17·31-s − 7.57e15·33-s − 8.26e17·35-s − 9.00e17·37-s − 6.77e17·39-s − 1.81e18·41-s + 9.02e18·43-s − 2.97e18·45-s + 2.64e18·47-s + 4.27e19·49-s + 6.46e18·51-s + 1.27e20·53-s + 3.38e18·55-s + ⋯
L(s)  = 1  − 0.697·3-s + 0.875·5-s − 1.65·7-s − 0.330·9-s + 0.0374·11-s + 0.489·13-s − 0.610·15-s − 0.213·17-s − 0.753·19-s + 1.15·21-s − 0.821·23-s − 1.84·25-s − 0.621·27-s − 2.17·29-s − 1.44·31-s − 0.0261·33-s − 1.44·35-s − 0.832·37-s − 0.341·39-s − 0.516·41-s + 1.48·43-s − 0.289·45-s + 0.156·47-s + 1.56·49-s + 0.149·51-s + 1.88·53-s + 0.0327·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+23/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-1$
Analytic conductor: \(19284.0\)
Root analytic conductor: \(5.17844\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 512,\ (\ :23/2, 23/2, 23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 + 7924 p^{3} T + 948725057 p^{4} T^{2} + 2085160918904 p^{9} T^{3} + 948725057 p^{27} T^{4} + 7924 p^{49} T^{5} + p^{69} T^{6} \)
5$S_4\times C_2$ \( 1 - 95628618 T + 6231350534237367 p T^{2} - \)\(28\!\cdots\!44\)\( p^{4} T^{3} + 6231350534237367 p^{24} T^{4} - 95628618 p^{46} T^{5} + p^{69} T^{6} \)
7$S_4\times C_2$ \( 1 + 8647912920 T + 653348641610189109 p^{2} T^{2} + \)\(33\!\cdots\!88\)\( p^{4} T^{3} + 653348641610189109 p^{25} T^{4} + 8647912920 p^{46} T^{5} + p^{69} T^{6} \)
11$S_4\times C_2$ \( 1 - 3220082436 p T + \)\(37\!\cdots\!97\)\( p^{2} T^{2} + \)\(63\!\cdots\!04\)\( p^{3} T^{3} + \)\(37\!\cdots\!97\)\( p^{25} T^{4} - 3220082436 p^{47} T^{5} + p^{69} T^{6} \)
13$S_4\times C_2$ \( 1 - 3164858452338 T + \)\(98\!\cdots\!11\)\( T^{2} - \)\(21\!\cdots\!36\)\( p T^{3} + \)\(98\!\cdots\!11\)\( p^{23} T^{4} - 3164858452338 p^{46} T^{5} + p^{69} T^{6} \)
17$S_4\times C_2$ \( 1 + 30233487828906 T + \)\(33\!\cdots\!67\)\( p T^{2} + \)\(39\!\cdots\!04\)\( p^{2} T^{3} + \)\(33\!\cdots\!67\)\( p^{24} T^{4} + 30233487828906 p^{46} T^{5} + p^{69} T^{6} \)
19$S_4\times C_2$ \( 1 + 20144988652644 p T + \)\(18\!\cdots\!41\)\( p^{2} T^{2} + \)\(28\!\cdots\!68\)\( p^{3} T^{3} + \)\(18\!\cdots\!41\)\( p^{25} T^{4} + 20144988652644 p^{47} T^{5} + p^{69} T^{6} \)
23$S_4\times C_2$ \( 1 + 3754416434163720 T + \)\(55\!\cdots\!33\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{3} + \)\(55\!\cdots\!33\)\( p^{23} T^{4} + 3754416434163720 p^{46} T^{5} + p^{69} T^{6} \)
29$S_4\times C_2$ \( 1 + 142892073311612862 T + \)\(18\!\cdots\!03\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!03\)\( p^{23} T^{4} + 142892073311612862 p^{46} T^{5} + p^{69} T^{6} \)
31$S_4\times C_2$ \( 1 + 204189369040807008 T + \)\(22\!\cdots\!33\)\( T^{2} - \)\(36\!\cdots\!44\)\( T^{3} + \)\(22\!\cdots\!33\)\( p^{23} T^{4} + 204189369040807008 p^{46} T^{5} + p^{69} T^{6} \)
37$S_4\times C_2$ \( 1 + 900641094521542422 T + \)\(33\!\cdots\!55\)\( T^{2} + \)\(32\!\cdots\!44\)\( T^{3} + \)\(33\!\cdots\!55\)\( p^{23} T^{4} + 900641094521542422 p^{46} T^{5} + p^{69} T^{6} \)
41$S_4\times C_2$ \( 1 + 1818448729363485042 T + \)\(12\!\cdots\!83\)\( T^{2} + \)\(75\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!83\)\( p^{23} T^{4} + 1818448729363485042 p^{46} T^{5} + p^{69} T^{6} \)
43$S_4\times C_2$ \( 1 - 9021528635440809420 T + \)\(90\!\cdots\!89\)\( T^{2} - \)\(42\!\cdots\!44\)\( T^{3} + \)\(90\!\cdots\!89\)\( p^{23} T^{4} - 9021528635440809420 p^{46} T^{5} + p^{69} T^{6} \)
47$S_4\times C_2$ \( 1 - 2644177465669087344 T + \)\(54\!\cdots\!81\)\( T^{2} - \)\(16\!\cdots\!16\)\( T^{3} + \)\(54\!\cdots\!81\)\( p^{23} T^{4} - 2644177465669087344 p^{46} T^{5} + p^{69} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(12\!\cdots\!54\)\( T + \)\(14\!\cdots\!03\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!03\)\( p^{23} T^{4} - \)\(12\!\cdots\!54\)\( p^{46} T^{5} + p^{69} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(74\!\cdots\!72\)\( T + \)\(34\!\cdots\!57\)\( T^{2} + \)\(94\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!57\)\( p^{23} T^{4} + \)\(74\!\cdots\!72\)\( p^{46} T^{5} + p^{69} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!62\)\( T + \)\(29\!\cdots\!43\)\( T^{2} + \)\(33\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!43\)\( p^{23} T^{4} + \)\(14\!\cdots\!62\)\( p^{46} T^{5} + p^{69} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(34\!\cdots\!28\)\( T + \)\(68\!\cdots\!29\)\( T^{2} + \)\(83\!\cdots\!04\)\( T^{3} + \)\(68\!\cdots\!29\)\( p^{23} T^{4} + \)\(34\!\cdots\!28\)\( p^{46} T^{5} + p^{69} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(40\!\cdots\!04\)\( T + \)\(14\!\cdots\!57\)\( T^{2} + \)\(30\!\cdots\!28\)\( T^{3} + \)\(14\!\cdots\!57\)\( p^{23} T^{4} + \)\(40\!\cdots\!04\)\( p^{46} T^{5} + p^{69} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(47\!\cdots\!90\)\( T + \)\(26\!\cdots\!23\)\( T^{2} - \)\(70\!\cdots\!72\)\( T^{3} + \)\(26\!\cdots\!23\)\( p^{23} T^{4} - \)\(47\!\cdots\!90\)\( p^{46} T^{5} + p^{69} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!28\)\( T + \)\(13\!\cdots\!73\)\( T^{2} - \)\(97\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!73\)\( p^{23} T^{4} - \)\(11\!\cdots\!28\)\( p^{46} T^{5} + p^{69} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(52\!\cdots\!36\)\( T + \)\(25\!\cdots\!81\)\( T^{2} - \)\(43\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!81\)\( p^{23} T^{4} + \)\(52\!\cdots\!36\)\( p^{46} T^{5} + p^{69} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(60\!\cdots\!26\)\( T + \)\(46\!\cdots\!91\)\( T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + \)\(46\!\cdots\!91\)\( p^{23} T^{4} + \)\(60\!\cdots\!26\)\( p^{46} T^{5} + p^{69} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(23\!\cdots\!34\)\( T + \)\(32\!\cdots\!71\)\( T^{2} + \)\(27\!\cdots\!16\)\( T^{3} + \)\(32\!\cdots\!71\)\( p^{23} T^{4} + \)\(23\!\cdots\!34\)\( p^{46} T^{5} + p^{69} T^{6} \)
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

−15.09939033309893883868358581475, −14.05225909907329262157755306806, −13.59572452201746085591125148885, −13.39229671775951288574836396720, −12.87184891258049384133492809063, −12.14946864444288184488189567580, −11.96848583375276439158193300236, −11.01063200835838902252696153898, −10.79524661974056312355186155720, −10.14926655185282999030615685866, −9.521333393595047162042737082394, −9.236811105282582652158344170786, −8.813520752673417876205738253510, −7.57189456182293364434945750553, −7.50094524633731962764719798148, −6.42466379102561216966283980141, −6.10621489963350830003614920012, −5.69275095569615358046547172310, −5.48023367535566271763754827417, −4.15551091387888452174319835526, −3.85560170941735166394106130503, −3.25143665535516040844785771045, −2.40879168467817439754597202072, −1.87429326647951696333603913423, −1.42610746650890802960986163753, 0, 0, 0, 1.42610746650890802960986163753, 1.87429326647951696333603913423, 2.40879168467817439754597202072, 3.25143665535516040844785771045, 3.85560170941735166394106130503, 4.15551091387888452174319835526, 5.48023367535566271763754827417, 5.69275095569615358046547172310, 6.10621489963350830003614920012, 6.42466379102561216966283980141, 7.50094524633731962764719798148, 7.57189456182293364434945750553, 8.813520752673417876205738253510, 9.236811105282582652158344170786, 9.521333393595047162042737082394, 10.14926655185282999030615685866, 10.79524661974056312355186155720, 11.01063200835838902252696153898, 11.96848583375276439158193300236, 12.14946864444288184488189567580, 12.87184891258049384133492809063, 13.39229671775951288574836396720, 13.59572452201746085591125148885, 14.05225909907329262157755306806, 15.09939033309893883868358581475

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