Properties

Label 6-1520e3-1.1-c1e3-0-2
Degree $6$
Conductor $3511808000$
Sign $1$
Analytic cond. $1787.97$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 7-s − 2·9-s + 5·13-s + 3·15-s + 5·17-s − 3·19-s + 21-s + 23-s + 6·25-s − 3·27-s + 17·29-s − 2·31-s + 3·35-s + 8·37-s + 5·39-s + 6·41-s + 10·43-s − 6·45-s − 4·47-s − 8·49-s + 5·51-s + 5·53-s − 3·57-s − 15·59-s + 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 0.377·7-s − 2/3·9-s + 1.38·13-s + 0.774·15-s + 1.21·17-s − 0.688·19-s + 0.218·21-s + 0.208·23-s + 6/5·25-s − 0.577·27-s + 3.15·29-s − 0.359·31-s + 0.507·35-s + 1.31·37-s + 0.800·39-s + 0.937·41-s + 1.52·43-s − 0.894·45-s − 0.583·47-s − 8/7·49-s + 0.700·51-s + 0.686·53-s − 0.397·57-s − 1.95·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(1787.97\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.564080476\)
\(L(\frac12)\) \(\approx\) \(7.564080476\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 - T + p T^{2} - 2 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 9 T^{2} + 2 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 - 5 T + 41 T^{2} - 126 T^{3} + 41 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 5 T + 47 T^{2} - 166 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 37 T^{2} + 18 T^{3} + 37 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 17 T + 171 T^{2} - 1110 T^{3} + 171 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 45 T^{2} - 4 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 137 T^{2} - 828 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 504 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 117 T^{2} - 582 T^{3} + 117 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 15 T + 149 T^{2} + 986 T^{3} + 149 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 3 T + 23 T^{2} - 650 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 21 T^{2} + 456 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 3 T + 119 T^{2} - 10 p T^{3} + 119 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 213 T^{2} + 284 T^{3} + 213 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 233 T^{2} - 1628 T^{3} + 233 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 187 T^{2} + 1072 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.608736615591084490492559729062, −7.977850034518973652271147128097, −7.971444700519335383311694286707, −7.907383327387767197198710348340, −7.39631650848874463642914827090, −6.96388599860259865490311383984, −6.70313351124936683446670931635, −6.32543172730690716256767398146, −6.14235486214750531307888602846, −6.12575812990272592151877704280, −5.60489610051275255889642719841, −5.46480609328795584388998748613, −5.10623204682097692601510458475, −4.65917345370058172650245295272, −4.35704139143883124490320642831, −4.31283939726720098525076730932, −3.63028567603728988654050298155, −3.26553310273189336483224958201, −3.06860030542496379244206288580, −2.76465589888172096731859452112, −2.31612204161527165908495480690, −2.06081549124745001629732123672, −1.51960481786494023859307073891, −0.936674243037242236417885398167, −0.861758686219750748911599781021, 0.861758686219750748911599781021, 0.936674243037242236417885398167, 1.51960481786494023859307073891, 2.06081549124745001629732123672, 2.31612204161527165908495480690, 2.76465589888172096731859452112, 3.06860030542496379244206288580, 3.26553310273189336483224958201, 3.63028567603728988654050298155, 4.31283939726720098525076730932, 4.35704139143883124490320642831, 4.65917345370058172650245295272, 5.10623204682097692601510458475, 5.46480609328795584388998748613, 5.60489610051275255889642719841, 6.12575812990272592151877704280, 6.14235486214750531307888602846, 6.32543172730690716256767398146, 6.70313351124936683446670931635, 6.96388599860259865490311383984, 7.39631650848874463642914827090, 7.907383327387767197198710348340, 7.971444700519335383311694286707, 7.977850034518973652271147128097, 8.608736615591084490492559729062

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