Properties

Degree $6$
Conductor $182284263$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s − 3·5-s + 3·7-s + 9·10-s − 6·11-s − 3·13-s − 9·14-s − 3·16-s − 6·17-s − 3·19-s − 9·20-s + 18·22-s − 12·23-s − 6·25-s + 9·26-s + 9·28-s − 9·29-s − 3·31-s + 6·32-s + 18·34-s − 9·35-s − 3·37-s + 9·38-s − 3·43-s − 18·44-s + 36·46-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 1.34·5-s + 1.13·7-s + 2.84·10-s − 1.80·11-s − 0.832·13-s − 2.40·14-s − 3/4·16-s − 1.45·17-s − 0.688·19-s − 2.01·20-s + 3.83·22-s − 2.50·23-s − 6/5·25-s + 1.76·26-s + 1.70·28-s − 1.67·29-s − 0.538·31-s + 1.06·32-s + 3.08·34-s − 1.52·35-s − 0.493·37-s + 1.45·38-s − 0.457·43-s − 2.71·44-s + 5.30·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{12} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{567} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{12} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good2$A_4\times C_2$ \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 6 T + 42 T^{2} + 135 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 + 3 T + 6 T^{2} - 29 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 3 T + 51 T^{2} + 97 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 12 T + 96 T^{2} + 549 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 9 T + 51 T^{2} + 189 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 114 T^{2} + 9 T^{3} + 114 p T^{4} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 18 T + 294 T^{2} - 2997 T^{3} + 294 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.10230894140699732046751167454, −9.572310937334592632146804226006, −9.554802214366261132376443105118, −8.983654195024787525592099519702, −8.760614867739603285440135304412, −8.577801434617660734396194093914, −8.245839697179878784779077894000, −7.913953997702209406289956428163, −7.78014400988830651791169088806, −7.71982549934738127554574527712, −7.22294042911838076464286174298, −7.13735710448351875663360313624, −6.50057767771042358522708295707, −5.97690938482399114685884158899, −5.74036438185373714314048704932, −5.46011383056055644730047054974, −4.84680165485764198067841520818, −4.53132672795117656662485945350, −4.37809598544036147863262519473, −3.80709206323046140474176002180, −3.69179883501881382005233149482, −2.76887134811980147821937033308, −2.38454120518082410602072645616, −1.85017468245371177253221244307, −1.67718686743966778762820678462, 0, 0, 0, 1.67718686743966778762820678462, 1.85017468245371177253221244307, 2.38454120518082410602072645616, 2.76887134811980147821937033308, 3.69179883501881382005233149482, 3.80709206323046140474176002180, 4.37809598544036147863262519473, 4.53132672795117656662485945350, 4.84680165485764198067841520818, 5.46011383056055644730047054974, 5.74036438185373714314048704932, 5.97690938482399114685884158899, 6.50057767771042358522708295707, 7.13735710448351875663360313624, 7.22294042911838076464286174298, 7.71982549934738127554574527712, 7.78014400988830651791169088806, 7.913953997702209406289956428163, 8.245839697179878784779077894000, 8.577801434617660734396194093914, 8.760614867739603285440135304412, 8.983654195024787525592099519702, 9.554802214366261132376443105118, 9.572310937334592632146804226006, 10.10230894140699732046751167454

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