Properties

Label 6-4851e3-1.1-c1e3-0-0
Degree $6$
Conductor $114154707051$
Sign $1$
Analytic cond. $58119.9$
Root an. cond. $6.22377$
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  − 4-s − 2·5-s + 3·8-s + 3·11-s + 11·13-s − 16-s − 3·17-s + 11·19-s + 2·20-s − 12·23-s − 4·25-s + 9·29-s + 3·31-s − 6·32-s − 4·37-s − 6·40-s − 5·41-s + 2·43-s − 3·44-s − 3·47-s − 11·52-s − 17·53-s − 6·55-s + 8·59-s + 24·61-s + 4·64-s − 22·65-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.904·11-s + 3.05·13-s − 1/4·16-s − 0.727·17-s + 2.52·19-s + 0.447·20-s − 2.50·23-s − 4/5·25-s + 1.67·29-s + 0.538·31-s − 1.06·32-s − 0.657·37-s − 0.948·40-s − 0.780·41-s + 0.304·43-s − 0.452·44-s − 0.437·47-s − 1.52·52-s − 2.33·53-s − 0.809·55-s + 1.04·59-s + 3.07·61-s + 1/2·64-s − 2.72·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(58119.9\)
Root analytic conductor: \(6.22377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{6} \cdot 7^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.391126980\)
\(L(\frac12)\) \(\approx\) \(4.391126980\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + T^{2} - 3 T^{3} + p T^{4} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 2 T + 8 T^{2} + 3 p T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 11 T + 75 T^{2} - 321 T^{3} + 75 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 3 T + 49 T^{2} + 95 T^{3} + 49 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 11 T + 77 T^{2} - p^{2} T^{3} + 77 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 12 T + 112 T^{2} + 599 T^{3} + 112 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 9 T + 67 T^{2} - 469 T^{3} + 67 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 49 T^{2} - 79 T^{3} + 49 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 75 T^{2} + 144 T^{3} + 75 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 5 T + 43 T^{2} + 301 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T + 104 T^{2} - 131 T^{3} + 104 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 3 T + 139 T^{2} + 275 T^{3} + 139 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 17 T + 233 T^{2} + 1823 T^{3} + 233 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 20 T^{2} + 379 T^{3} + 20 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 24 T + 355 T^{2} - 3304 T^{3} + 355 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 16 T + 269 T^{2} + 2216 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 7 T + 127 T^{2} + 575 T^{3} + 127 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 20 T + 244 T^{2} - 2295 T^{3} + 244 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 3 T + 199 T^{2} - 333 T^{3} + 199 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 11 T + 265 T^{2} + 1823 T^{3} + 265 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - T + 259 T^{2} - 175 T^{3} + 259 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 9 T + 279 T^{2} + 1699 T^{3} + 279 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.21279011455746323061411562326, −7.00988017298061158092343809400, −6.98517184990413893678516334225, −6.44874384327187820095265962024, −6.40382845255132285352377608037, −6.22887840491009776233946266571, −5.88676750696864865766974140166, −5.42552200247530533444261274832, −5.42129758730369450071366520549, −5.21438128436112609285710543649, −4.64187369408277660756196863364, −4.47817248206204272974536277921, −4.18842889087189281204322866383, −4.01349874302529494242384755989, −3.74964828316096338371615920835, −3.70298252841855573449015592285, −3.19086913292068307772041663155, −3.15353508231980236838488783745, −2.61813292620469963610271910035, −2.14976817219960933533198551941, −1.57339361260023822181313854175, −1.55435385167247344933819242208, −1.36907692918477724386975595162, −0.64472012682959174738343616310, −0.51694091230656924347063307889, 0.51694091230656924347063307889, 0.64472012682959174738343616310, 1.36907692918477724386975595162, 1.55435385167247344933819242208, 1.57339361260023822181313854175, 2.14976817219960933533198551941, 2.61813292620469963610271910035, 3.15353508231980236838488783745, 3.19086913292068307772041663155, 3.70298252841855573449015592285, 3.74964828316096338371615920835, 4.01349874302529494242384755989, 4.18842889087189281204322866383, 4.47817248206204272974536277921, 4.64187369408277660756196863364, 5.21438128436112609285710543649, 5.42129758730369450071366520549, 5.42552200247530533444261274832, 5.88676750696864865766974140166, 6.22887840491009776233946266571, 6.40382845255132285352377608037, 6.44874384327187820095265962024, 6.98517184990413893678516334225, 7.00988017298061158092343809400, 7.21279011455746323061411562326

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