Properties

Label 6-4026e3-1.1-c1e3-0-0
Degree $6$
Conductor $65256129576$
Sign $-1$
Analytic cond. $33224.0$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 6·4-s − 3·5-s − 9·6-s − 2·7-s + 10·8-s + 6·9-s − 9·10-s + 3·11-s − 18·12-s − 7·13-s − 6·14-s + 9·15-s + 15·16-s + 3·17-s + 18·18-s + 2·19-s − 18·20-s + 6·21-s + 9·22-s − 4·23-s − 30·24-s − 2·25-s − 21·26-s − 10·27-s − 12·28-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s − 0.755·7-s + 3.53·8-s + 2·9-s − 2.84·10-s + 0.904·11-s − 5.19·12-s − 1.94·13-s − 1.60·14-s + 2.32·15-s + 15/4·16-s + 0.727·17-s + 4.24·18-s + 0.458·19-s − 4.02·20-s + 1.30·21-s + 1.91·22-s − 0.834·23-s − 6.12·24-s − 2/5·25-s − 4.11·26-s − 1.92·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3}\)
Sign: $-1$
Analytic conductor: \(33224.0\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 11^{3} \cdot 61^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
61$C_1$ \( ( 1 + T )^{3} \)
good5$S_4\times C_2$ \( 1 + 3 T + 11 T^{2} + 21 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 7 T + 29 T^{2} + 85 T^{3} + 29 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 - 2 T + 32 T^{2} - 107 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 48 T^{2} + 103 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 8 T + 76 T^{2} + 437 T^{3} + 76 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 17 T + 175 T^{2} + 37 p T^{3} + 175 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 14 T + 160 T^{2} + 1045 T^{3} + 160 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 5 T + 66 T^{2} + 437 T^{3} + 66 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 122 T^{2} - 3 T^{3} + 122 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 5 T + 59 T^{2} + 303 T^{3} + 59 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 2 p T^{2} - 545 T^{3} + 2 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 13 T + 127 T^{2} - 733 T^{3} + 127 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 17 T + 283 T^{2} - 2371 T^{3} + 283 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 17 T + 295 T^{2} + 2507 T^{3} + 295 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 220 T^{2} - 1349 T^{3} + 220 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 7 T + 227 T^{2} + 1009 T^{3} + 227 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 14 T + 168 T^{2} + 2063 T^{3} + 168 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 23 T + 261 T^{2} + 2357 T^{3} + 261 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 7 T + 161 T^{2} + 709 T^{3} + 161 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.73178993757805577729691324672, −7.23854089455714257154126515543, −7.20750477721326401825918189796, −7.04483914453183360312576505885, −6.75825759159854996255530425037, −6.65250903595048022935882940782, −6.41428380425225101450061747214, −5.75435964542530765566829335260, −5.70358881871356866927879269626, −5.59633433107810369190922571473, −5.31833251900842588515353794314, −5.15821540174972337854022879804, −5.01511305046454481447815315754, −4.39975430718677059068053503375, −4.30624063203648590166672939610, −3.99703069694419138957501151629, −3.71917841496805893677609226403, −3.56497943546119986380325937466, −3.56221318610584867510695219703, −3.02675117602580223500398283999, −2.41891838768116842667420966337, −2.39822626927580260040702055062, −1.70460147229285180361580201030, −1.59603448868073353924323482731, −1.21635534741111314794764307812, 0, 0, 0, 1.21635534741111314794764307812, 1.59603448868073353924323482731, 1.70460147229285180361580201030, 2.39822626927580260040702055062, 2.41891838768116842667420966337, 3.02675117602580223500398283999, 3.56221318610584867510695219703, 3.56497943546119986380325937466, 3.71917841496805893677609226403, 3.99703069694419138957501151629, 4.30624063203648590166672939610, 4.39975430718677059068053503375, 5.01511305046454481447815315754, 5.15821540174972337854022879804, 5.31833251900842588515353794314, 5.59633433107810369190922571473, 5.70358881871356866927879269626, 5.75435964542530765566829335260, 6.41428380425225101450061747214, 6.65250903595048022935882940782, 6.75825759159854996255530425037, 7.04483914453183360312576505885, 7.20750477721326401825918189796, 7.23854089455714257154126515543, 7.73178993757805577729691324672

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