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Properties

Label 6-2e12-1.1-c23e3-0-0
Degree $6$
Conductor $4096$
Sign $1$
Analytic cond. $154272.$
Root an. cond. $7.32343$
Motivic weight $23$
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  − 3.27e4·3-s + 3.14e7·5-s − 9.93e8·7-s − 1.39e11·9-s + 2.34e10·11-s + 2.01e12·13-s − 1.02e12·15-s − 2.16e12·17-s − 3.12e14·19-s + 3.24e13·21-s − 4.77e15·23-s − 7.48e15·25-s − 1.44e16·27-s + 1.72e17·29-s − 4.24e17·31-s − 7.66e14·33-s − 3.12e16·35-s + 2.54e18·37-s − 6.60e16·39-s + 2.76e18·41-s + 5.96e18·43-s − 4.40e18·45-s + 5.25e19·47-s − 5.86e19·49-s + 7.06e16·51-s + 1.56e19·53-s + 7.37e17·55-s + ⋯
L(s)  = 1  − 0.106·3-s + 0.288·5-s − 0.189·7-s − 1.48·9-s + 0.0247·11-s + 0.312·13-s − 0.0307·15-s − 0.0152·17-s − 0.614·19-s + 0.0202·21-s − 1.04·23-s − 0.627·25-s − 0.499·27-s + 2.62·29-s − 3.00·31-s − 0.00264·33-s − 0.0547·35-s + 2.35·37-s − 0.0333·39-s + 0.783·41-s + 0.978·43-s − 0.428·45-s + 3.10·47-s − 2.14·49-s + 0.00162·51-s + 0.232·53-s + 0.00714·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+23/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(154272.\)
Root analytic conductor: \(7.32343\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 4096,\ (\ :23/2, 23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(1.706492375\)
\(L(\frac12)\) \(\approx\) \(1.706492375\)
\(L(\frac{25}{2})\) 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$S_4\times C_2$ \( 1 + 32708 T + 15672779593 p^{2} T^{2} + 10806152860616 p^{7} T^{3} + 15672779593 p^{25} T^{4} + 32708 p^{46} T^{5} + p^{69} T^{6} \)
5$S_4\times C_2$ \( 1 - 1259226 p^{2} T + 338935636667019 p^{2} T^{2} - \)\(42\!\cdots\!64\)\( p^{5} T^{3} + 338935636667019 p^{25} T^{4} - 1259226 p^{48} T^{5} + p^{69} T^{6} \)
7$S_4\times C_2$ \( 1 + 141860760 p T + 1217330227180247349 p^{2} T^{2} + \)\(14\!\cdots\!48\)\( p^{4} T^{3} + 1217330227180247349 p^{25} T^{4} + 141860760 p^{47} T^{5} + p^{69} T^{6} \)
11$S_4\times C_2$ \( 1 - 2131047804 p T + \)\(51\!\cdots\!07\)\( p T^{2} + \)\(38\!\cdots\!56\)\( p^{2} T^{3} + \)\(51\!\cdots\!07\)\( p^{24} T^{4} - 2131047804 p^{47} T^{5} + p^{69} T^{6} \)
13$S_4\times C_2$ \( 1 - 2019379246962 T + \)\(33\!\cdots\!07\)\( p T^{2} - \)\(43\!\cdots\!88\)\( p^{2} T^{3} + \)\(33\!\cdots\!07\)\( p^{24} T^{4} - 2019379246962 p^{46} T^{5} + p^{69} T^{6} \)
17$S_4\times C_2$ \( 1 + 2160517821354 T + \)\(13\!\cdots\!27\)\( p T^{2} + \)\(51\!\cdots\!36\)\( p^{2} T^{3} + \)\(13\!\cdots\!27\)\( p^{24} T^{4} + 2160517821354 p^{46} T^{5} + p^{69} T^{6} \)
19$S_4\times C_2$ \( 1 + 312191787410964 T + \)\(17\!\cdots\!79\)\( p T^{2} - \)\(38\!\cdots\!92\)\( p^{2} T^{3} + \)\(17\!\cdots\!79\)\( p^{24} T^{4} + 312191787410964 p^{46} T^{5} + p^{69} T^{6} \)
23$S_4\times C_2$ \( 1 + 4776103684514040 T + \)\(31\!\cdots\!13\)\( T^{2} + \)\(19\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!13\)\( p^{23} T^{4} + 4776103684514040 p^{46} T^{5} + p^{69} T^{6} \)
29$S_4\times C_2$ \( 1 - 172150256810092098 T + \)\(20\!\cdots\!03\)\( T^{2} - \)\(15\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!03\)\( p^{23} T^{4} - 172150256810092098 p^{46} T^{5} + p^{69} T^{6} \)
31$S_4\times C_2$ \( 1 + 424442363135920032 T + \)\(11\!\cdots\!53\)\( T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!53\)\( p^{23} T^{4} + 424442363135920032 p^{46} T^{5} + p^{69} T^{6} \)
37$S_4\times C_2$ \( 1 - 2543857445080432362 T + \)\(52\!\cdots\!95\)\( T^{2} - \)\(61\!\cdots\!04\)\( T^{3} + \)\(52\!\cdots\!95\)\( p^{23} T^{4} - 2543857445080432362 p^{46} T^{5} + p^{69} T^{6} \)
41$S_4\times C_2$ \( 1 - 2761653172426159758 T + \)\(33\!\cdots\!83\)\( T^{2} - \)\(69\!\cdots\!52\)\( T^{3} + \)\(33\!\cdots\!83\)\( p^{23} T^{4} - 2761653172426159758 p^{46} T^{5} + p^{69} T^{6} \)
43$S_4\times C_2$ \( 1 - 5964473505832923060 T + \)\(41\!\cdots\!89\)\( T^{2} - \)\(49\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!89\)\( p^{23} T^{4} - 5964473505832923060 p^{46} T^{5} + p^{69} T^{6} \)
47$S_4\times C_2$ \( 1 - 52578892233811303824 T + \)\(17\!\cdots\!61\)\( T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!61\)\( p^{23} T^{4} - 52578892233811303824 p^{46} T^{5} + p^{69} T^{6} \)
53$S_4\times C_2$ \( 1 - 15683594666449490106 T + \)\(67\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!43\)\( p^{23} T^{4} - 15683594666449490106 p^{46} T^{5} + p^{69} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!52\)\( T - \)\(31\!\cdots\!23\)\( T^{2} + \)\(36\!\cdots\!80\)\( T^{3} - \)\(31\!\cdots\!23\)\( p^{23} T^{4} - \)\(10\!\cdots\!52\)\( p^{46} T^{5} + p^{69} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(12\!\cdots\!98\)\( T + \)\(84\!\cdots\!83\)\( T^{2} - \)\(34\!\cdots\!76\)\( T^{3} + \)\(84\!\cdots\!83\)\( p^{23} T^{4} - \)\(12\!\cdots\!98\)\( p^{46} T^{5} + p^{69} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(80\!\cdots\!12\)\( T + \)\(20\!\cdots\!49\)\( T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!49\)\( p^{23} T^{4} - \)\(80\!\cdots\!12\)\( p^{46} T^{5} + p^{69} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(32\!\cdots\!24\)\( T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(20\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!57\)\( p^{23} T^{4} - \)\(32\!\cdots\!24\)\( p^{46} T^{5} + p^{69} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(25\!\cdots\!30\)\( T + \)\(22\!\cdots\!23\)\( T^{2} - \)\(35\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!23\)\( p^{23} T^{4} - \)\(25\!\cdots\!30\)\( p^{46} T^{5} + p^{69} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(17\!\cdots\!12\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(16\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!93\)\( p^{23} T^{4} - \)\(17\!\cdots\!12\)\( p^{46} T^{5} + p^{69} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(15\!\cdots\!04\)\( T + \)\(38\!\cdots\!41\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(38\!\cdots\!41\)\( p^{23} T^{4} - \)\(15\!\cdots\!04\)\( p^{46} T^{5} + p^{69} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!06\)\( T - \)\(37\!\cdots\!89\)\( T^{2} - \)\(22\!\cdots\!48\)\( T^{3} - \)\(37\!\cdots\!89\)\( p^{23} T^{4} + \)\(13\!\cdots\!06\)\( p^{46} T^{5} + p^{69} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!66\)\( T + \)\(98\!\cdots\!71\)\( T^{2} - \)\(23\!\cdots\!16\)\( T^{3} + \)\(98\!\cdots\!71\)\( p^{23} T^{4} + \)\(12\!\cdots\!66\)\( p^{46} T^{5} + p^{69} T^{6} \)
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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

−12.24568128244365548803352744159, −11.37504494944828124547995213739, −11.08837964933455262220551857746, −10.92596856702214220519219601667, −10.22910566989800139030292443762, −9.550194003815126768990861531803, −9.437343522565014067807455490989, −8.847383068054939377793382559397, −8.205806843425221516808454777909, −8.063902192582090160800014473556, −7.50923332867011883277858026227, −6.59235675492909348729885415518, −6.50653075932645208603905936688, −5.81856978781255548344215586789, −5.51927791811514047390996285072, −5.16104194246477660674447101612, −4.26766970509539386416334053664, −3.77195550604131399950799700244, −3.64398855234585664120293951390, −2.56321964118566970646992675143, −2.35156542686021608242830305163, −2.20380358493201778867980771090, −1.09399667589722974520135022475, −0.811951196299998065081727421981, −0.25687485062335511450238145431, 0.25687485062335511450238145431, 0.811951196299998065081727421981, 1.09399667589722974520135022475, 2.20380358493201778867980771090, 2.35156542686021608242830305163, 2.56321964118566970646992675143, 3.64398855234585664120293951390, 3.77195550604131399950799700244, 4.26766970509539386416334053664, 5.16104194246477660674447101612, 5.51927791811514047390996285072, 5.81856978781255548344215586789, 6.50653075932645208603905936688, 6.59235675492909348729885415518, 7.50923332867011883277858026227, 8.063902192582090160800014473556, 8.205806843425221516808454777909, 8.847383068054939377793382559397, 9.437343522565014067807455490989, 9.550194003815126768990861531803, 10.22910566989800139030292443762, 10.92596856702214220519219601667, 11.08837964933455262220551857746, 11.37504494944828124547995213739, 12.24568128244365548803352744159

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