Properties

Label 6-7728e3-1.1-c1e3-0-2
Degree $6$
Conductor $461531492352$
Sign $1$
Analytic cond. $234980.$
Root an. cond. $7.85546$
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  − 3·3-s − 3·5-s + 3·7-s + 6·9-s + 2·11-s + 5·13-s + 9·15-s − 4·17-s + 14·19-s − 9·21-s − 3·23-s + 25-s − 10·27-s − 4·29-s + 6·31-s − 6·33-s − 9·35-s − 6·37-s − 15·39-s + 10·41-s + 21·43-s − 18·45-s + 2·47-s + 6·49-s + 12·51-s − 13·53-s − 6·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s + 1.13·7-s + 2·9-s + 0.603·11-s + 1.38·13-s + 2.32·15-s − 0.970·17-s + 3.21·19-s − 1.96·21-s − 0.625·23-s + 1/5·25-s − 1.92·27-s − 0.742·29-s + 1.07·31-s − 1.04·33-s − 1.52·35-s − 0.986·37-s − 2.40·39-s + 1.56·41-s + 3.20·43-s − 2.68·45-s + 0.291·47-s + 6/7·49-s + 1.68·51-s − 1.78·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(234980.\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.666521102\)
\(L(\frac12)\) \(\approx\) \(2.666521102\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good5$S_4\times C_2$ \( 1 + 3 T + 8 T^{2} + 2 p T^{3} + 8 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 24 T^{2} - 30 T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 5 T + 40 T^{2} - 128 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 32 T^{2} + 86 T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 14 T + 112 T^{2} - 578 T^{3} + 112 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 68 T^{2} + 182 T^{3} + 68 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 6 T + 50 T^{2} - 212 T^{3} + 50 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 6 T + 68 T^{2} + 400 T^{3} + 68 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 10 T + 146 T^{2} - 830 T^{3} + 146 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 21 T + 212 T^{2} - 1504 T^{3} + 212 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 113 T^{2} - 124 T^{3} + 113 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 13 T + 162 T^{2} + 1362 T^{3} + 162 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 5 T + 86 T^{2} + 406 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 17 T + 218 T^{2} + 1878 T^{3} + 218 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 13 T + 158 T^{2} - 1416 T^{3} + 158 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + T + 206 T^{2} + 134 T^{3} + 206 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 134 T^{2} - 326 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 16 T + 312 T^{2} - 2632 T^{3} + 312 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 168 T^{2} + 1018 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 11 T + 124 T^{2} + 636 T^{3} + 124 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 288 T^{2} - 1526 T^{3} + 288 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.04546963904112918211790020336, −6.62470663386164135049755548349, −6.35737638883119377242503889056, −6.24064871062307287237729858510, −6.03877269345023593295704421968, −5.80975771277602586128946132360, −5.49219960696436655379563910641, −5.23812221943613600430524385851, −5.21630668360250801978224669582, −4.80167731719864991315616531794, −4.43914081267001439839062008761, −4.39549268481482109913874999577, −4.22589069066422677463146776073, −3.76421634019797170499959965063, −3.65047376048885859974614144152, −3.53826642114445110750898210500, −3.01411131596091871693899868618, −2.69426963811074424665354093681, −2.46432325897463466208635351480, −1.71629752938478360838545203394, −1.71158656745884287684929701431, −1.35921471796453170117873633247, −0.867859080206768168506541000624, −0.75929154463493432190187908974, −0.40564453456331370986961568661, 0.40564453456331370986961568661, 0.75929154463493432190187908974, 0.867859080206768168506541000624, 1.35921471796453170117873633247, 1.71158656745884287684929701431, 1.71629752938478360838545203394, 2.46432325897463466208635351480, 2.69426963811074424665354093681, 3.01411131596091871693899868618, 3.53826642114445110750898210500, 3.65047376048885859974614144152, 3.76421634019797170499959965063, 4.22589069066422677463146776073, 4.39549268481482109913874999577, 4.43914081267001439839062008761, 4.80167731719864991315616531794, 5.21630668360250801978224669582, 5.23812221943613600430524385851, 5.49219960696436655379563910641, 5.80975771277602586128946132360, 6.03877269345023593295704421968, 6.24064871062307287237729858510, 6.35737638883119377242503889056, 6.62470663386164135049755548349, 7.04546963904112918211790020336

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