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Properties

Label 6-392e3-1.1-c3e3-0-1
Degree $6$
Conductor $60236288$
Sign $1$
Analytic cond. $12372.4$
Root an. cond. $4.80923$
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  + 3-s + 13·5-s − 15·9-s − 11·11-s + 70·13-s + 13·15-s + 97·17-s − 81·19-s + 191·23-s − 97·25-s + 14·27-s − 162·29-s + 597·31-s − 11·33-s + 217·37-s + 70·39-s + 698·41-s − 308·43-s − 195·45-s + 139·47-s + 97·51-s + 197·53-s − 143·55-s − 81·57-s + 353·59-s + 449·61-s + 910·65-s + ⋯
L(s)  = 1  + 0.192·3-s + 1.16·5-s − 5/9·9-s − 0.301·11-s + 1.49·13-s + 0.223·15-s + 1.38·17-s − 0.978·19-s + 1.73·23-s − 0.775·25-s + 0.0997·27-s − 1.03·29-s + 3.45·31-s − 0.0580·33-s + 0.964·37-s + 0.287·39-s + 2.65·41-s − 1.09·43-s − 0.645·45-s + 0.431·47-s + 0.266·51-s + 0.510·53-s − 0.350·55-s − 0.188·57-s + 0.778·59-s + 0.942·61-s + 1.73·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60236288 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60236288 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(60236288\)    =    \(2^{9} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(12372.4\)
Root analytic conductor: \(4.80923\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 60236288,\ (\ :3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.670519731\)
\(L(\frac12)\) \(\approx\) \(7.670519731\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$S_4\times C_2$ \( 1 - T + 16 T^{2} - 5 p^{2} T^{3} + 16 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \)
5$S_4\times C_2$ \( 1 - 13 T + 266 T^{2} - 2001 T^{3} + 266 p^{3} T^{4} - 13 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + p T + 3416 T^{2} + 21183 T^{3} + 3416 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 70 T + 5435 T^{2} - 289828 T^{3} + 5435 p^{3} T^{4} - 70 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 97 T + 8494 T^{2} - 366565 T^{3} + 8494 p^{3} T^{4} - 97 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 81 T + 10248 T^{2} + 987781 T^{3} + 10248 p^{3} T^{4} + 81 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 191 T + 19204 T^{2} - 1155515 T^{3} + 19204 p^{3} T^{4} - 191 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 162 T + 65451 T^{2} + 6577740 T^{3} + 65451 p^{3} T^{4} + 162 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 597 T + 192972 T^{2} - 40233593 T^{3} + 192972 p^{3} T^{4} - 597 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 217 T + 74466 T^{2} - 6873029 T^{3} + 74466 p^{3} T^{4} - 217 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 698 T + 297255 T^{2} - 89748524 T^{3} + 297255 p^{3} T^{4} - 698 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 308 T + 230633 T^{2} + 49015160 T^{3} + 230633 p^{3} T^{4} + 308 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 139 T + 197052 T^{2} - 10153063 T^{3} + 197052 p^{3} T^{4} - 139 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 197 T + 387082 T^{2} - 61572377 T^{3} + 387082 p^{3} T^{4} - 197 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 353 T + 315688 T^{2} - 107706461 T^{3} + 315688 p^{3} T^{4} - 353 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 449 T + 362602 T^{2} - 213771085 T^{3} + 362602 p^{3} T^{4} - 449 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 519 T + 545904 T^{2} + 290868187 T^{3} + 545904 p^{3} T^{4} + 519 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 224 T + 694277 T^{2} - 152792640 T^{3} + 694277 p^{3} T^{4} - 224 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1701 T + 1858686 T^{2} - 1396157665 T^{3} + 1858686 p^{3} T^{4} - 1701 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 1143 T + 1126572 T^{2} - 829925491 T^{3} + 1126572 p^{3} T^{4} - 1143 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 1380 T + 2190993 T^{2} + 1582929496 T^{3} + 2190993 p^{3} T^{4} + 1380 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1749 T + 2664750 T^{2} - 2310361041 T^{3} + 2664750 p^{3} T^{4} - 1749 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 2602 T + 4832863 T^{2} - 5278291436 T^{3} + 4832863 p^{3} T^{4} - 2602 p^{6} T^{5} + p^{9} T^{6} \)
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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

−9.819843087933681791398416035342, −9.260672010112167600927393211003, −8.981753805551296539776093391556, −8.971461616826234862215585028841, −8.338000291476023308371374033641, −8.128693252101459951506237591607, −7.962239717736107307550122154708, −7.41965080932842015287594919472, −7.28336264090903957026005972392, −6.48910498118744096874950536519, −6.28051943445724506820673729744, −6.15529181742905057424271640437, −5.97412150101933017076717341729, −5.34798627786237645938081474040, −5.08046562231734337133129301430, −4.81558867666371840807579152243, −4.16315605692927769936345620564, −3.70995855420754359608698849097, −3.56712428130351366756286795829, −2.76465450362503978854700807388, −2.57380241362482216801834163328, −2.22763586076472509831000822703, −1.46763261897674756920723698631, −0.853406078351368893529275579045, −0.75861408699973835244799407150, 0.75861408699973835244799407150, 0.853406078351368893529275579045, 1.46763261897674756920723698631, 2.22763586076472509831000822703, 2.57380241362482216801834163328, 2.76465450362503978854700807388, 3.56712428130351366756286795829, 3.70995855420754359608698849097, 4.16315605692927769936345620564, 4.81558867666371840807579152243, 5.08046562231734337133129301430, 5.34798627786237645938081474040, 5.97412150101933017076717341729, 6.15529181742905057424271640437, 6.28051943445724506820673729744, 6.48910498118744096874950536519, 7.28336264090903957026005972392, 7.41965080932842015287594919472, 7.962239717736107307550122154708, 8.128693252101459951506237591607, 8.338000291476023308371374033641, 8.971461616826234862215585028841, 8.981753805551296539776093391556, 9.260672010112167600927393211003, 9.819843087933681791398416035342

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