Properties

Degree $6$
Conductor $2662500456$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s + 2·5-s + 3·7-s + 10·8-s + 6·10-s + 3·11-s + 9·14-s + 15·16-s + 2·17-s + 10·19-s + 12·20-s + 9·22-s + 2·23-s − 25-s + 18·28-s − 6·29-s + 21·32-s + 6·34-s + 6·35-s + 10·37-s + 30·38-s + 20·40-s − 2·41-s + 14·43-s + 18·44-s + 6·46-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 0.894·5-s + 1.13·7-s + 3.53·8-s + 1.89·10-s + 0.904·11-s + 2.40·14-s + 15/4·16-s + 0.485·17-s + 2.29·19-s + 2.68·20-s + 1.91·22-s + 0.417·23-s − 1/5·25-s + 3.40·28-s − 1.11·29-s + 3.71·32-s + 1.02·34-s + 1.01·35-s + 1.64·37-s + 4.86·38-s + 3.16·40-s − 0.312·41-s + 2.13·43-s + 2.71·44-s + 0.884·46-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(26.27586764\)
\(L(\frac12)\) \(\approx\) \(26.27586764\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 - 2 T + p T^{2} - 8 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 5 T^{2} - 16 T^{3} - 5 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 41 T^{2} - 56 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 10 T + 51 T^{2} - 192 T^{3} + 51 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + 13 T^{2} - 44 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 6 T + 55 T^{2} + 252 T^{3} + 55 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 9 T^{2} - 128 T^{3} - 9 p T^{4} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 99 T^{2} - 588 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 113 T^{2} + 152 T^{3} + 113 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 14 T + 73 T^{2} - 228 T^{3} + 73 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 10 T + 103 T^{2} + 544 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 179 T^{2} - 1412 T^{3} + 179 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 41 T^{2} - 208 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 8 T + 43 T^{2} + 192 T^{3} + 43 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 4 T + 149 T^{2} - 472 T^{3} + 149 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 2 T + 53 T^{2} - 580 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 14 T + 273 T^{2} + 2088 T^{3} + 273 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 8 T + 201 T^{2} - 1120 T^{3} + 201 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 20 T + 371 T^{2} + 3536 T^{3} + 371 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 16 T + 295 T^{2} + 2704 T^{3} + 295 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 18 T + 223 T^{2} - 2524 T^{3} + 223 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.552777985617149827046325058896, −8.111583538292374115460263718703, −7.68712141052455163926656827676, −7.57757689610496881944711607465, −7.27101098499960963634692509281, −7.17395496757360296755173465586, −6.86783298906984883531605130672, −6.29791579561805203836526474112, −6.06133985051057111849455336419, −5.93753404362871426181355861379, −5.62572784863285190930442472674, −5.42194327190271186074624051041, −5.12258704213171560060491740868, −4.81467675683958722081657223521, −4.50583889468581131997651863894, −4.28350688314488872518318196562, −3.75698826294922420259469282777, −3.58630503822586028230338006360, −3.38001615177652481692593974125, −2.72746456133807310065222655570, −2.39503063058201151847913837817, −2.37487282676375360395858562462, −1.49587078039262481950061262573, −1.35060704710633302988938480794, −1.03809843908470062371467227493, 1.03809843908470062371467227493, 1.35060704710633302988938480794, 1.49587078039262481950061262573, 2.37487282676375360395858562462, 2.39503063058201151847913837817, 2.72746456133807310065222655570, 3.38001615177652481692593974125, 3.58630503822586028230338006360, 3.75698826294922420259469282777, 4.28350688314488872518318196562, 4.50583889468581131997651863894, 4.81467675683958722081657223521, 5.12258704213171560060491740868, 5.42194327190271186074624051041, 5.62572784863285190930442472674, 5.93753404362871426181355861379, 6.06133985051057111849455336419, 6.29791579561805203836526474112, 6.86783298906984883531605130672, 7.17395496757360296755173465586, 7.27101098499960963634692509281, 7.57757689610496881944711607465, 7.68712141052455163926656827676, 8.111583538292374115460263718703, 8.552777985617149827046325058896

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