Properties

Label 6-4840e3-1.1-c1e3-0-5
Degree $6$
Conductor $113379904000$
Sign $-1$
Analytic cond. $57725.4$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 7-s − 3·15-s − 7·17-s − 3·19-s − 21-s − 4·23-s + 6·25-s − 27-s − 5·29-s − 17·31-s + 3·35-s + 37-s + 2·41-s − 16·43-s − 6·47-s − 12·49-s + 7·51-s + 53-s + 3·57-s − 4·59-s − 11·61-s − 16·67-s + 4·69-s + 71-s + 22·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s − 0.774·15-s − 1.69·17-s − 0.688·19-s − 0.218·21-s − 0.834·23-s + 6/5·25-s − 0.192·27-s − 0.928·29-s − 3.05·31-s + 0.507·35-s + 0.164·37-s + 0.312·41-s − 2.43·43-s − 0.875·47-s − 1.71·49-s + 0.980·51-s + 0.137·53-s + 0.397·57-s − 0.520·59-s − 1.40·61-s − 1.95·67-s + 0.481·69-s + 0.118·71-s + 2.57·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{3} \cdot 11^{6}\)
Sign: $-1$
Analytic conductor: \(57725.4\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + T^{2} + 2 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 13 T^{2} - 10 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T^{2} - 64 T^{3} - p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 7 T + 59 T^{2} + 230 T^{3} + 59 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 17 T^{2} + 130 T^{3} + 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 71 T^{2} + 226 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 17 T + 165 T^{2} + 1070 T^{3} + 165 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 91 T^{2} - 132 T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 16 T + 189 T^{2} + 1360 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 6 T + 113 T^{2} + 556 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 135 T^{2} - 126 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 81 T^{2} + 600 T^{3} + 81 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 11 T + 95 T^{2} + 6 p T^{3} + 95 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 16 T + 165 T^{2} + 1168 T^{3} + 165 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - T + 85 T^{2} - 654 T^{3} + 85 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 22 T + 355 T^{2} - 3388 T^{3} + 355 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{3} \)
83$S_4\times C_2$ \( 1 + 21 T^{2} + 880 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 3 T + 83 T^{2} + 1138 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
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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

−7.79885204692911857871196854730, −7.23939641472957166396732713135, −7.23614537708720825789409377286, −6.93828290655216915099283382995, −6.59495093761316008673455740518, −6.40365679468772869254277846044, −6.29389802488028478714301337554, −5.87488907698797742121153585402, −5.72350191801035797690362032090, −5.67760441039495144810101383270, −5.14967785059423325891284483978, −5.00508981389497106931216591502, −4.81762696078143412693884849513, −4.42204591444375277392075002498, −4.38984414196363608212795204488, −3.78029323219123561974239947564, −3.66496910739223453797290688296, −3.39532540018809211318598093508, −2.99001427910649527202317956989, −2.61603869146978843024056451333, −2.28287811080297923439640962905, −1.92790031856497923850970461919, −1.82316574374174274975689127338, −1.41373210106233268813919583471, −1.33206469780861092655692292895, 0, 0, 0, 1.33206469780861092655692292895, 1.41373210106233268813919583471, 1.82316574374174274975689127338, 1.92790031856497923850970461919, 2.28287811080297923439640962905, 2.61603869146978843024056451333, 2.99001427910649527202317956989, 3.39532540018809211318598093508, 3.66496910739223453797290688296, 3.78029323219123561974239947564, 4.38984414196363608212795204488, 4.42204591444375277392075002498, 4.81762696078143412693884849513, 5.00508981389497106931216591502, 5.14967785059423325891284483978, 5.67760441039495144810101383270, 5.72350191801035797690362032090, 5.87488907698797742121153585402, 6.29389802488028478714301337554, 6.40365679468772869254277846044, 6.59495093761316008673455740518, 6.93828290655216915099283382995, 7.23614537708720825789409377286, 7.23939641472957166396732713135, 7.79885204692911857871196854730

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