Properties

Label 6-6864e3-1.1-c1e3-0-0
Degree $6$
Conductor $323393900544$
Sign $1$
Analytic cond. $164650.$
Root an. cond. $7.40333$
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  − 3·3-s − 4·5-s + 6·7-s + 6·9-s − 3·11-s + 3·13-s + 12·15-s + 6·19-s − 18·21-s + 3·25-s − 10·27-s − 2·29-s + 14·31-s + 9·33-s − 24·35-s − 8·37-s − 9·39-s − 4·41-s − 6·43-s − 24·45-s + 10·47-s + 11·49-s − 14·53-s + 12·55-s − 18·57-s − 4·59-s + 4·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s + 2.26·7-s + 2·9-s − 0.904·11-s + 0.832·13-s + 3.09·15-s + 1.37·19-s − 3.92·21-s + 3/5·25-s − 1.92·27-s − 0.371·29-s + 2.51·31-s + 1.56·33-s − 4.05·35-s − 1.31·37-s − 1.44·39-s − 0.624·41-s − 0.914·43-s − 3.57·45-s + 1.45·47-s + 11/7·49-s − 1.92·53-s + 1.61·55-s − 2.38·57-s − 0.520·59-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(164650.\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.813954702\)
\(L(\frac12)\) \(\approx\) \(1.813954702\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 + T )^{3} \)
13$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 + 4 T + 13 T^{2} + 26 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 6 T + 25 T^{2} - 72 T^{3} + 25 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 3 T^{2} - 74 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 61 T^{2} - 224 T^{3} + 61 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 3 T^{2} + 108 T^{3} - 3 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 83 T^{2} + 110 T^{3} + 83 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 139 T^{2} - 910 T^{3} + 139 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 T + 63 T^{2} + 304 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 59 T^{2} + 460 T^{3} + 59 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 93 T^{2} + 354 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 10 T + 145 T^{2} - 932 T^{3} + 145 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 14 T + 203 T^{2} + 1508 T^{3} + 203 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 105 T^{2} + 616 T^{3} + 105 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 4 T + 87 T^{2} - 632 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 22 T + 251 T^{2} - 2086 T^{3} + 251 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 193 T^{2} - 828 T^{3} + 193 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T + 207 T^{2} - 548 T^{3} + 207 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 22 T + 373 T^{2} - 3650 T^{3} + 373 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 16 T + 257 T^{2} + 2640 T^{3} + 257 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 173 T^{2} + 614 T^{3} + 173 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 24 T + 451 T^{2} + 4944 T^{3} + 451 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19791480038701363399291175378, −6.66234422622519743953474196914, −6.58454085803582791423679186860, −6.57272088860453203776393299414, −5.95286009348591616409175772402, −5.71935454302311262707885985748, −5.61567897880384939575105516275, −5.21598595998499094436677325749, −5.05949708288702806537251648419, −5.03804923039050650054704908871, −4.63534042877687707171738311504, −4.47413620749754394392723396472, −4.31999920985390292643428866573, −3.74203375663625967250108933134, −3.72507466828502982881287949825, −3.63516285062417893401090891861, −2.96989125042364779647030262805, −2.84564189069411439058468649375, −2.44483609880182626691745167249, −1.76379012993947609705281228783, −1.71913193089937421582851178747, −1.50738121371879159353853034489, −0.891932050236332853054426865322, −0.66726691544710570416462432806, −0.37186469345270579816772238051, 0.37186469345270579816772238051, 0.66726691544710570416462432806, 0.891932050236332853054426865322, 1.50738121371879159353853034489, 1.71913193089937421582851178747, 1.76379012993947609705281228783, 2.44483609880182626691745167249, 2.84564189069411439058468649375, 2.96989125042364779647030262805, 3.63516285062417893401090891861, 3.72507466828502982881287949825, 3.74203375663625967250108933134, 4.31999920985390292643428866573, 4.47413620749754394392723396472, 4.63534042877687707171738311504, 5.03804923039050650054704908871, 5.05949708288702806537251648419, 5.21598595998499094436677325749, 5.61567897880384939575105516275, 5.71935454302311262707885985748, 5.95286009348591616409175772402, 6.57272088860453203776393299414, 6.58454085803582791423679186860, 6.66234422622519743953474196914, 7.19791480038701363399291175378

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