Properties

Label 6-2160e3-1.1-c3e3-0-14
Degree $6$
Conductor $10077696000$
Sign $-1$
Analytic cond. $2.06994\times 10^{6}$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·5-s − 6·7-s − 12·11-s + 18·13-s − 21·17-s − 57·19-s − 87·23-s + 150·25-s + 138·29-s − 117·31-s − 90·35-s + 150·37-s − 180·43-s − 684·47-s − 537·49-s + 87·53-s − 180·55-s − 714·59-s − 513·61-s + 270·65-s + 174·67-s − 768·71-s − 252·73-s + 72·77-s − 207·79-s − 1.68e3·83-s − 315·85-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.323·7-s − 0.328·11-s + 0.384·13-s − 0.299·17-s − 0.688·19-s − 0.788·23-s + 6/5·25-s + 0.883·29-s − 0.677·31-s − 0.434·35-s + 0.666·37-s − 0.638·43-s − 2.12·47-s − 1.56·49-s + 0.225·53-s − 0.441·55-s − 1.57·59-s − 1.07·61-s + 0.515·65-s + 0.317·67-s − 1.28·71-s − 0.404·73-s + 0.106·77-s − 0.294·79-s − 2.23·83-s − 0.401·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{9} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(2.06994\times 10^{6}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 6 T + 573 T^{2} + 20 p^{3} T^{3} + 573 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 12 T + 2733 T^{2} + 11024 T^{3} + 2733 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 18 T + 5391 T^{2} - 55708 T^{3} + 5391 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 21 T + 7458 T^{2} + 253565 T^{3} + 7458 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 p T + 8592 T^{2} + 1114405 T^{3} + 8592 p^{3} T^{4} + 3 p^{7} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 87 T + 28512 T^{2} + 1461623 T^{3} + 28512 p^{3} T^{4} + 87 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 138 T + 20103 T^{2} - 123916 p T^{3} + 20103 p^{3} T^{4} - 138 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 117 T + 20988 T^{2} - 2423207 T^{3} + 20988 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 150 T + 95907 T^{2} - 17945796 T^{3} + 95907 p^{3} T^{4} - 150 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 68055 T^{2} + 17332488 T^{3} + 68055 p^{3} T^{4} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 180 T + 22293 T^{2} - 3328688 T^{3} + 22293 p^{3} T^{4} + 180 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 684 T + 347853 T^{2} + 110757224 T^{3} + 347853 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 87 T + 277926 T^{2} + 2139897 T^{3} + 277926 p^{3} T^{4} - 87 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 714 T + 669321 T^{2} + 272637508 T^{3} + 669321 p^{3} T^{4} + 714 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 513 T + 466794 T^{2} + 175503629 T^{3} + 466794 p^{3} T^{4} + 513 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 174 T + 677229 T^{2} - 48960124 T^{3} + 677229 p^{3} T^{4} - 174 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 768 T + 869433 T^{2} + 382859112 T^{3} + 869433 p^{3} T^{4} + 768 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 252 T + 967647 T^{2} + 141584400 T^{3} + 967647 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 207 T + 981660 T^{2} + 306571995 T^{3} + 981660 p^{3} T^{4} + 207 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 1689 T + 2368140 T^{2} + 1880402401 T^{3} + 2368140 p^{3} T^{4} + 1689 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 312 T + 1492215 T^{2} - 319203960 T^{3} + 1492215 p^{3} T^{4} - 312 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1080 T + 1410147 T^{2} + 2105654384 T^{3} + 1410147 p^{3} T^{4} + 1080 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.335708803474657741869336035588, −7.67682805048906370127508881385, −7.54565259561919279666686949237, −7.52007660086296131986211088141, −6.73041916278856542134557643235, −6.66004149579822095053248657075, −6.65019177973329127314784165489, −6.14535112315621334335229627728, −6.02843975436681544320287897534, −5.82806547027103508837548573601, −5.48133968217765639533537091414, −5.05235395478381034850180219421, −4.96167930500241211309875740163, −4.44994517994404549184798042231, −4.43266241555921281444754296037, −4.09842801858865825149822117094, −3.31258977008535191507790073022, −3.28309569594611300921282391616, −3.25471155193961193987421052162, −2.52590540362649521589829466787, −2.22816492338724674413679441763, −2.20934440912494615610265893096, −1.39760805951814472794312813056, −1.39349413646942090878245744678, −1.16571036471750443774664437546, 0, 0, 0, 1.16571036471750443774664437546, 1.39349413646942090878245744678, 1.39760805951814472794312813056, 2.20934440912494615610265893096, 2.22816492338724674413679441763, 2.52590540362649521589829466787, 3.25471155193961193987421052162, 3.28309569594611300921282391616, 3.31258977008535191507790073022, 4.09842801858865825149822117094, 4.43266241555921281444754296037, 4.44994517994404549184798042231, 4.96167930500241211309875740163, 5.05235395478381034850180219421, 5.48133968217765639533537091414, 5.82806547027103508837548573601, 6.02843975436681544320287897534, 6.14535112315621334335229627728, 6.65019177973329127314784165489, 6.66004149579822095053248657075, 6.73041916278856542134557643235, 7.52007660086296131986211088141, 7.54565259561919279666686949237, 7.67682805048906370127508881385, 8.335708803474657741869336035588

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