Properties

Degree $6$
Conductor $104088305664$
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  + 3·3-s + 6·9-s − 3·13-s + 6·17-s + 3·19-s − 6·23-s − 6·25-s + 10·27-s + 12·29-s + 3·31-s + 3·37-s − 9·39-s − 6·41-s + 15·43-s + 12·47-s + 18·51-s + 6·53-s + 9·57-s + 12·59-s − 18·61-s + 9·67-s − 18·69-s + 33·73-s − 18·75-s − 27·79-s + 15·81-s + 18·83-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 0.832·13-s + 1.45·17-s + 0.688·19-s − 1.25·23-s − 6/5·25-s + 1.92·27-s + 2.22·29-s + 0.538·31-s + 0.493·37-s − 1.44·39-s − 0.937·41-s + 2.28·43-s + 1.75·47-s + 2.52·51-s + 0.824·53-s + 1.19·57-s + 1.56·59-s − 2.30·61-s + 1.09·67-s − 2.16·69-s + 3.86·73-s − 2.07·75-s − 3.03·79-s + 5/3·81-s + 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{15} \cdot 3^{3} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{15} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.76931781\)
\(L(\frac12)\) \(\approx\) \(12.76931781\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{3} \)
7 \( 1 \)
good5$S_4\times C_2$ \( 1 + 6 T^{2} + 4 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T^{2} + 38 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 6 T + 27 T^{2} - 108 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 T + 21 T^{2} - 2 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 12 T + 126 T^{2} - 728 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 72 T^{2} - 139 T^{3} + 72 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 3 T + 27 T^{2} + 146 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 15 T + 177 T^{2} - 1318 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 12 T + 153 T^{2} - 1016 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 6 T + 78 T^{2} - 802 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 198 T^{2} - 1334 T^{3} + 198 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 18 T + 195 T^{2} + 1644 T^{3} + 195 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 9 T + 189 T^{2} - 1042 T^{3} + 189 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 33 T + 555 T^{2} - 5814 T^{3} + 555 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 27 T + 432 T^{2} + 4619 T^{3} + 432 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 18 T + 234 T^{2} - 2040 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 12 T + 279 T^{2} + 2024 T^{3} + 279 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} \)
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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

−7.52855679030438035013983537739, −7.32235997848384174768214349395, −6.96613580577173477164782515090, −6.66181228486540900968061592231, −6.48938379199950495942802554401, −5.99610128016405089436380660165, −5.88011220658887777047232268547, −5.73634222958864683396539915222, −5.41572699543653049373586273671, −4.97994090233378409864230624513, −4.64062627788031100277872170587, −4.57098221543713095288888371649, −4.39923729214898377460747563937, −3.75271320774962777977482074465, −3.68110904109015502664387347680, −3.62536066849341689170256069394, −3.15986202700257577522614561196, −2.82838676027020833433008019278, −2.52004236532419179863488110333, −2.41740677076200962223455075500, −1.96304359688206906527789484334, −1.85756170627488036768507741764, −1.15631719000364565764745986505, −0.74199297978788934558945188527, −0.70275787593495915748312305404, 0.70275787593495915748312305404, 0.74199297978788934558945188527, 1.15631719000364565764745986505, 1.85756170627488036768507741764, 1.96304359688206906527789484334, 2.41740677076200962223455075500, 2.52004236532419179863488110333, 2.82838676027020833433008019278, 3.15986202700257577522614561196, 3.62536066849341689170256069394, 3.68110904109015502664387347680, 3.75271320774962777977482074465, 4.39923729214898377460747563937, 4.57098221543713095288888371649, 4.64062627788031100277872170587, 4.97994090233378409864230624513, 5.41572699543653049373586273671, 5.73634222958864683396539915222, 5.88011220658887777047232268547, 5.99610128016405089436380660165, 6.48938379199950495942802554401, 6.66181228486540900968061592231, 6.96613580577173477164782515090, 7.32235997848384174768214349395, 7.52855679030438035013983537739

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