Properties

Label 6-532e3-1.1-c1e3-0-0
Degree $6$
Conductor $150568768$
Sign $1$
Analytic cond. $76.6595$
Root an. cond. $2.06107$
Motivic weight $1$
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 + 2·5-s − 3·7-s − 9-s − 3·11-s + 8·13-s − 2·15-s + 9·17-s − 3·19-s + 3·21-s + 4·23-s − 2·25-s + 5·27-s + 11·29-s − 5·31-s + 3·33-s − 6·35-s + 4·37-s − 8·39-s + 11·41-s − 4·43-s − 2·45-s − 2·47-s + 6·49-s − 9·51-s + 9·53-s − 6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 1.13·7-s − 1/3·9-s − 0.904·11-s + 2.21·13-s − 0.516·15-s + 2.18·17-s − 0.688·19-s + 0.654·21-s + 0.834·23-s − 2/5·25-s + 0.962·27-s + 2.04·29-s − 0.898·31-s + 0.522·33-s − 1.01·35-s + 0.657·37-s − 1.28·39-s + 1.71·41-s − 0.609·43-s − 0.298·45-s − 0.291·47-s + 6/7·49-s − 1.26·51-s + 1.23·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 7^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(76.6595\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{6} \cdot 7^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.895970982\)
\(L(\frac12)\) \(\approx\) \(1.895970982\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + 2 T^{2} - 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 2 T + 6 T^{2} - 6 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 26 T^{2} + 46 T^{3} + 26 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 50 T^{2} - 192 T^{3} + 50 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 9 T + 4 p T^{2} - 292 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 64 T^{2} - 180 T^{3} + 64 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 11 T + 102 T^{2} - 636 T^{3} + 102 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 5 T + 76 T^{2} + 314 T^{3} + 76 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 92 T^{2} - 246 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 11 T + 110 T^{2} - 622 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 2 T + 66 T^{2} + 372 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 9 T + 120 T^{2} - 632 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 132 T^{2} + 1308 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 26 T^{2} + 42 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 158 T^{2} - 58 T^{3} + 158 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 7 T + 204 T^{2} + 1062 T^{3} + 204 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 18 T + 335 T^{2} + 3268 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.949763854125872228922766102331, −9.173684892032731244241915386119, −9.123785993389711056716324861753, −8.983681525750068249055621304982, −8.422758858790219811209059391082, −8.255644367519764528697889914668, −7.87876783057187559915424389150, −7.53006651195627836331587804110, −7.22620686671880440743459859965, −6.78165107694721299597808291444, −6.30796350637534742595841829157, −6.17080486661833610394981287929, −6.02292060836695184786515810754, −5.70251231015803247666193158640, −5.23112615924531473989453653101, −5.22157330715317075106525899720, −4.35626440572185263704594242378, −4.22143293068117386592167793388, −3.69577256103269503615127567635, −3.07824342109271019014623291957, −2.93622564705540637267829896199, −2.74825576966717284429858589035, −1.74947587594220826752490724642, −1.27823511515292654918125295076, −0.67232681957611485259072177045, 0.67232681957611485259072177045, 1.27823511515292654918125295076, 1.74947587594220826752490724642, 2.74825576966717284429858589035, 2.93622564705540637267829896199, 3.07824342109271019014623291957, 3.69577256103269503615127567635, 4.22143293068117386592167793388, 4.35626440572185263704594242378, 5.22157330715317075106525899720, 5.23112615924531473989453653101, 5.70251231015803247666193158640, 6.02292060836695184786515810754, 6.17080486661833610394981287929, 6.30796350637534742595841829157, 6.78165107694721299597808291444, 7.22620686671880440743459859965, 7.53006651195627836331587804110, 7.87876783057187559915424389150, 8.255644367519764528697889914668, 8.422758858790219811209059391082, 8.983681525750068249055621304982, 9.123785993389711056716324861753, 9.173684892032731244241915386119, 9.949763854125872228922766102331

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