Properties

Label 6-91e6-1.1-c1e3-0-4
Degree $6$
Conductor $567869252041$
Sign $1$
Analytic cond. $289121.$
Root an. cond. $8.13167$
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  + 2·2-s − 4·3-s + 2·4-s + 5·5-s − 8·6-s + 2·8-s + 9·9-s + 10·10-s + 4·11-s − 8·12-s − 20·15-s − 4·17-s + 18·18-s + 7·19-s + 10·20-s + 8·22-s + 23-s − 8·24-s + 7·25-s − 18·27-s − 7·29-s − 40·30-s − 3·31-s − 16·33-s − 8·34-s + 18·36-s + 10·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 4-s + 2.23·5-s − 3.26·6-s + 0.707·8-s + 3·9-s + 3.16·10-s + 1.20·11-s − 2.30·12-s − 5.16·15-s − 0.970·17-s + 4.24·18-s + 1.60·19-s + 2.23·20-s + 1.70·22-s + 0.208·23-s − 1.63·24-s + 7/5·25-s − 3.46·27-s − 1.29·29-s − 7.30·30-s − 0.538·31-s − 2.78·33-s − 1.37·34-s + 3·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(289121.\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8281} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 7^{6} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.89111562\)
\(L(\frac12)\) \(\approx\) \(10.89111562\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13 \( 1 \)
good2$S_4\times C_2$ \( 1 - p T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 + 4 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - p T + 18 T^{2} - 43 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 86 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 49 T^{2} + 122 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 7 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 28 T^{2} - 89 T^{3} + 28 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 3 T + 52 T^{2} + 137 T^{3} + 52 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 119 T^{2} - 658 T^{3} + 119 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 17 T + 230 T^{2} - 1745 T^{3} + 230 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 13 T + 198 T^{2} - 1369 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 22 T + 321 T^{2} - 2848 T^{3} + 321 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 24 T + 343 T^{2} - 3152 T^{3} + 343 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 14 T + 165 T^{2} - 1228 T^{3} + 165 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 4 T + 169 T^{2} + 374 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 5 T + 831 T^{3} - 5 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - T + 168 T^{2} - 257 T^{3} + 168 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 23 T + 376 T^{2} - 4021 T^{3} + 376 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 11 T + 212 T^{2} + 1979 T^{3} + 212 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 3 T + 220 T^{2} + 575 T^{3} + 220 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.67350865829358908360512895579, −6.48786136644011322375480923770, −6.24108440303215173256404088765, −5.94715506852979863448943507247, −5.93562357359485896955969275169, −5.82916513184496964069658724978, −5.49595902772674697995909568971, −5.20390189038762578538344343010, −5.17290072840982956152108519001, −5.08425461536798465618062388197, −4.67507656251326487871301936774, −4.24944451574368659750592856101, −4.02542808054566654085491915691, −3.90381647000755896503015727584, −3.71364271127611282802453082658, −3.63825766967085331467122271132, −2.66466018407331687136427332055, −2.64579642552097015045775589852, −2.22528494464595079400037718683, −2.09311181540657651544959481323, −2.05828685363559002453424967774, −1.22113651464751189382690369961, −1.05219104733623274491776326182, −0.908673617798597382957446926800, −0.46858713459258572921726944162, 0.46858713459258572921726944162, 0.908673617798597382957446926800, 1.05219104733623274491776326182, 1.22113651464751189382690369961, 2.05828685363559002453424967774, 2.09311181540657651544959481323, 2.22528494464595079400037718683, 2.64579642552097015045775589852, 2.66466018407331687136427332055, 3.63825766967085331467122271132, 3.71364271127611282802453082658, 3.90381647000755896503015727584, 4.02542808054566654085491915691, 4.24944451574368659750592856101, 4.67507656251326487871301936774, 5.08425461536798465618062388197, 5.17290072840982956152108519001, 5.20390189038762578538344343010, 5.49595902772674697995909568971, 5.82916513184496964069658724978, 5.93562357359485896955969275169, 5.94715506852979863448943507247, 6.24108440303215173256404088765, 6.48786136644011322375480923770, 6.67350865829358908360512895579

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