Properties

Label 6-6042e3-1.1-c1e3-0-2
Degree $6$
Conductor $220567826088$
Sign $-1$
Analytic cond. $112298.$
Root an. cond. $6.94590$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 3·3-s + 6·4-s − 7·5-s + 9·6-s − 7·7-s + 10·8-s + 6·9-s − 21·10-s + 5·11-s + 18·12-s + 13-s − 21·14-s − 21·15-s + 15·16-s − 5·17-s + 18·18-s + 3·19-s − 42·20-s − 21·21-s + 15·22-s + 23-s + 30·24-s + 20·25-s + 3·26-s + 10·27-s − 42·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s − 3.13·5-s + 3.67·6-s − 2.64·7-s + 3.53·8-s + 2·9-s − 6.64·10-s + 1.50·11-s + 5.19·12-s + 0.277·13-s − 5.61·14-s − 5.42·15-s + 15/4·16-s − 1.21·17-s + 4.24·18-s + 0.688·19-s − 9.39·20-s − 4.58·21-s + 3.19·22-s + 0.208·23-s + 6.12·24-s + 4·25-s + 0.588·26-s + 1.92·27-s − 7.93·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3}\)
Sign: $-1$
Analytic conductor: \(112298.\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
19$C_1$ \( ( 1 - T )^{3} \)
53$C_1$ \( ( 1 - T )^{3} \)
good5$A_4\times C_2$ \( 1 + 7 T + 29 T^{2} + 77 T^{3} + 29 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + p T + 5 p T^{2} + 15 p T^{3} + 5 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 5 T + 39 T^{2} - 111 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 - T + 23 T^{2} + 3 T^{3} + 23 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 5 T + 43 T^{2} + 129 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - T + 11 T^{2} - 59 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 13 T + 123 T^{2} + 949 T^{3} + 123 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 32 T^{2} - 7 p T^{3} + 32 p T^{4} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 9 T + 149 T^{2} - 787 T^{3} + 149 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 13 T + 132 T^{2} + 873 T^{3} + 132 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 12 T + 204 T^{2} + 1389 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 18 T + 284 T^{2} + 2377 T^{3} + 284 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 11 T + 127 T^{2} + 1193 T^{3} + 127 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 20 T + 281 T^{2} + 2848 T^{3} + 281 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 7 T + 121 T^{2} - 1225 T^{3} + 121 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 9 T + 75 T^{2} - 693 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + T + 163 T^{2} + 417 T^{3} + 163 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 4 T + 186 T^{2} - 879 T^{3} + 186 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 29 T + 555 T^{2} + 6395 T^{3} + 555 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.51866130728459759430848613945, −7.11093657970390347709241314004, −7.01045640436872247836337237529, −6.88386386970522783417810812127, −6.49004255498610055479225739186, −6.39871406668185415183564191371, −6.32843748995162096533150911746, −5.67451048980104445335048455907, −5.60979284745292626435828872958, −5.35360662843117230550185456922, −4.63748381785000333273376840690, −4.61171400516338912930635801631, −4.52966894594578414342000109186, −3.96329799818854712730788427566, −3.85827572131819567479539257335, −3.79205521637451613749694983098, −3.54607876840365650529555526302, −3.48919580073189489310266632415, −3.18733339309133600350587898166, −2.88484503184079908131815094623, −2.67393406657497866352544998172, −2.45686231198600464719346151502, −1.65040642445215873452624500379, −1.44340106237141818202854656246, −1.42608166270408133426410336063, 0, 0, 0, 1.42608166270408133426410336063, 1.44340106237141818202854656246, 1.65040642445215873452624500379, 2.45686231198600464719346151502, 2.67393406657497866352544998172, 2.88484503184079908131815094623, 3.18733339309133600350587898166, 3.48919580073189489310266632415, 3.54607876840365650529555526302, 3.79205521637451613749694983098, 3.85827572131819567479539257335, 3.96329799818854712730788427566, 4.52966894594578414342000109186, 4.61171400516338912930635801631, 4.63748381785000333273376840690, 5.35360662843117230550185456922, 5.60979284745292626435828872958, 5.67451048980104445335048455907, 6.32843748995162096533150911746, 6.39871406668185415183564191371, 6.49004255498610055479225739186, 6.88386386970522783417810812127, 7.01045640436872247836337237529, 7.11093657970390347709241314004, 7.51866130728459759430848613945

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