Properties

Label 6-3640e3-1.1-c1e3-0-0
Degree $6$
Conductor $48228544000$
Sign $1$
Analytic cond. $24554.7$
Root an. cond. $5.39124$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s + 3·7-s + 9-s − 3·13-s + 3·15-s + 17-s − 5·19-s − 3·21-s + 4·23-s + 6·25-s + 5·27-s − 9·29-s + 11·31-s − 9·35-s + 37-s + 3·39-s + 41-s − 10·43-s − 3·45-s + 8·47-s + 6·49-s − 51-s + 6·53-s + 5·57-s + 9·59-s − 30·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.14·19-s − 0.654·21-s + 0.834·23-s + 6/5·25-s + 0.962·27-s − 1.67·29-s + 1.97·31-s − 1.52·35-s + 0.164·37-s + 0.480·39-s + 0.156·41-s − 1.52·43-s − 0.447·45-s + 1.16·47-s + 6/7·49-s − 0.140·51-s + 0.824·53-s + 0.662·57-s + 1.17·59-s − 3.84·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(24554.7\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3640} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.364531958\)
\(L(\frac12)\) \(\approx\) \(2.364531958\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
13$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + T - 2 p T^{3} + p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 - T + 36 T^{2} - 16 T^{3} + 36 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 5 T + 50 T^{2} + 182 T^{3} + 50 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 33 T^{2} - 136 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 9 T + 72 T^{2} + 320 T^{3} + 72 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 11 T + 4 p T^{2} - 706 T^{3} + 4 p^{2} T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - T + 52 T^{2} + 112 T^{3} + 52 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - T + 108 T^{2} - 64 T^{3} + 108 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 125 T^{2} + 844 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 121 T^{2} - 656 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 - 9 T + 120 T^{2} - 1078 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 - T + 86 T^{2} - 466 T^{3} + 86 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 2 T + 177 T^{2} - 188 T^{3} + 177 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 75 T^{2} - 1004 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 5 T + 230 T^{2} - 782 T^{3} + 230 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 213 T^{2} + 616 T^{3} + 213 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + T + 116 T^{2} + 564 T^{3} + 116 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 4 T + 59 T^{2} - 1792 T^{3} + 59 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.54312165642481378372799501941, −7.49817768243142560319627084004, −7.02563063596333029396700723542, −6.89820170844058866871540719525, −6.59303621552810476118701737877, −6.28566543851752052196195069280, −6.27671680147623795093464811242, −5.63156341558429639577363175206, −5.38030660287290178254403216167, −5.36250824826443544376028398660, −4.99364018922852794547145236527, −4.57076375070863188575331311390, −4.47353853358568203516691275983, −4.27971420414855641776744818235, −4.14134494086035703470492591008, −3.70220862204438419341270941492, −3.11323311213912354113152463034, −3.03416869963661629802707739462, −2.93150573880115084134502196142, −2.20576192420505254849829142256, −1.98858395828053413216408193115, −1.68587159636540516011195790087, −1.14850584815261554197436153461, −0.58017825424418269792892065228, −0.51233293304449176811288634343, 0.51233293304449176811288634343, 0.58017825424418269792892065228, 1.14850584815261554197436153461, 1.68587159636540516011195790087, 1.98858395828053413216408193115, 2.20576192420505254849829142256, 2.93150573880115084134502196142, 3.03416869963661629802707739462, 3.11323311213912354113152463034, 3.70220862204438419341270941492, 4.14134494086035703470492591008, 4.27971420414855641776744818235, 4.47353853358568203516691275983, 4.57076375070863188575331311390, 4.99364018922852794547145236527, 5.36250824826443544376028398660, 5.38030660287290178254403216167, 5.63156341558429639577363175206, 6.27671680147623795093464811242, 6.28566543851752052196195069280, 6.59303621552810476118701737877, 6.89820170844058866871540719525, 7.02563063596333029396700723542, 7.49817768243142560319627084004, 7.54312165642481378372799501941

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