Properties

Label 6-1368e3-1.1-c3e3-0-0
Degree $6$
Conductor $2560108032$
Sign $1$
Analytic cond. $525843.$
Root an. cond. $8.98413$
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  − 7·5-s + 7·7-s − 103·11-s + 32·13-s − 11·17-s − 57·19-s − 316·23-s − 82·25-s + 138·29-s + 420·31-s − 49·35-s + 102·37-s + 370·41-s + 431·43-s + 199·47-s − 891·49-s + 308·53-s + 721·55-s + 188·59-s − 609·61-s − 224·65-s − 246·67-s + 954·71-s − 629·73-s − 721·77-s + 452·79-s + 780·83-s + ⋯
L(s)  = 1  − 0.626·5-s + 0.377·7-s − 2.82·11-s + 0.682·13-s − 0.156·17-s − 0.688·19-s − 2.86·23-s − 0.655·25-s + 0.883·29-s + 2.43·31-s − 0.236·35-s + 0.453·37-s + 1.40·41-s + 1.52·43-s + 0.617·47-s − 2.59·49-s + 0.798·53-s + 1.76·55-s + 0.414·59-s − 1.27·61-s − 0.427·65-s − 0.448·67-s + 1.59·71-s − 1.00·73-s − 1.06·77-s + 0.643·79-s + 1.03·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(525843.\)
Root analytic conductor: \(8.98413\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{6} \cdot 19^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.769015814\)
\(L(\frac12)\) \(\approx\) \(1.769015814\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{3} \)
good5$S_4\times C_2$ \( 1 + 7 T + 131 T^{2} + 1142 T^{3} + 131 p^{3} T^{4} + 7 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 - p T + 940 T^{2} - 4243 T^{3} + 940 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 103 T + 6205 T^{2} + 252030 T^{3} + 6205 p^{3} T^{4} + 103 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 32 T + 2914 T^{2} - 9188 T^{3} + 2914 p^{3} T^{4} - 32 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 11 T + 666 p T^{2} + 140259 T^{3} + 666 p^{4} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 316 T + 59648 T^{2} + 7447208 T^{3} + 59648 p^{3} T^{4} + 316 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 138 T + 73586 T^{2} - 6385598 T^{3} + 73586 p^{3} T^{4} - 138 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 420 T + 144957 T^{2} - 27360696 T^{3} + 144957 p^{3} T^{4} - 420 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 102 T - 10893 T^{2} + 5232188 T^{3} - 10893 p^{3} T^{4} - 102 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 370 T + 242923 T^{2} - 51708996 T^{3} + 242923 p^{3} T^{4} - 370 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 431 T + 218013 T^{2} - 63114554 T^{3} + 218013 p^{3} T^{4} - 431 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 199 T + 96357 T^{2} + 3984558 T^{3} + 96357 p^{3} T^{4} - 199 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 308 T + 105730 T^{2} - 98338872 T^{3} + 105730 p^{3} T^{4} - 308 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 188 T + 613942 T^{2} - 76145262 T^{3} + 613942 p^{3} T^{4} - 188 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 609 T + 637811 T^{2} + 225844310 T^{3} + 637811 p^{3} T^{4} + 609 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 246 T + 507842 T^{2} + 51728764 T^{3} + 507842 p^{3} T^{4} + 246 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 954 T + 904961 T^{2} - 445620740 T^{3} + 904961 p^{3} T^{4} - 954 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 629 T + 522206 T^{2} + 72331857 T^{3} + 522206 p^{3} T^{4} + 629 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 452 T + 451225 T^{2} + 155904568 T^{3} + 451225 p^{3} T^{4} - 452 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 780 T + 311213 T^{2} + 156799240 T^{3} + 311213 p^{3} T^{4} - 780 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1356 T + 2683643 T^{2} - 1983548248 T^{3} + 2683643 p^{3} T^{4} - 1356 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 548 T + 149095 T^{2} - 1035193144 T^{3} + 149095 p^{3} T^{4} + 548 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.223834607992527798035250090443, −7.81007820441396723987890900230, −7.80311875964689259767438035785, −7.60165612502379080435796559388, −7.18624896997145261462184048214, −6.63778334734036382536019789928, −6.48540704370522943084394513308, −6.04031598578913490798043405643, −5.92782654503080430395915149829, −5.81141440956033451756835764575, −5.27087304134275618165960456498, −4.95325963562381644586389083996, −4.69757155332745383256131538028, −4.32713675282019254256712809019, −4.07626648596919512053100121966, −4.06239483658531536916375386736, −3.17457306963062186392959062198, −3.07627925548040200749505196756, −2.83237459269324859353575245478, −2.13104078861082743545416752178, −2.10507117737632768662242998237, −1.87732773814116378343467387801, −0.77270677029144829245397450576, −0.75273871135936216228170881703, −0.26945740142061255147756475104, 0.26945740142061255147756475104, 0.75273871135936216228170881703, 0.77270677029144829245397450576, 1.87732773814116378343467387801, 2.10507117737632768662242998237, 2.13104078861082743545416752178, 2.83237459269324859353575245478, 3.07627925548040200749505196756, 3.17457306963062186392959062198, 4.06239483658531536916375386736, 4.07626648596919512053100121966, 4.32713675282019254256712809019, 4.69757155332745383256131538028, 4.95325963562381644586389083996, 5.27087304134275618165960456498, 5.81141440956033451756835764575, 5.92782654503080430395915149829, 6.04031598578913490798043405643, 6.48540704370522943084394513308, 6.63778334734036382536019789928, 7.18624896997145261462184048214, 7.60165612502379080435796559388, 7.80311875964689259767438035785, 7.81007820441396723987890900230, 8.223834607992527798035250090443

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