Properties

Label 6-3042e3-1.1-c1e3-0-2
Degree $6$
Conductor $28149950088$
Sign $1$
Analytic cond. $14332.0$
Root an. cond. $4.92853$
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  + 3·2-s + 6·4-s + 5-s − 9·7-s + 10·8-s + 3·10-s + 5·11-s − 27·14-s + 15·16-s + 8·17-s + 4·19-s + 6·20-s + 15·22-s + 2·25-s − 54·28-s − 11·29-s − 5·31-s + 21·32-s + 24·34-s − 9·35-s + 8·37-s + 12·38-s + 10·40-s + 2·41-s + 12·43-s + 30·44-s + 4·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 0.447·5-s − 3.40·7-s + 3.53·8-s + 0.948·10-s + 1.50·11-s − 7.21·14-s + 15/4·16-s + 1.94·17-s + 0.917·19-s + 1.34·20-s + 3.19·22-s + 2/5·25-s − 10.2·28-s − 2.04·29-s − 0.898·31-s + 3.71·32-s + 4.11·34-s − 1.52·35-s + 1.31·37-s + 1.94·38-s + 1.58·40-s + 0.312·41-s + 1.82·43-s + 4.52·44-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(14332.0\)
Root analytic conductor: \(4.92853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.98197898\)
\(L(\frac12)\) \(\approx\) \(13.98197898\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
3 \( 1 \)
13 \( 1 \)
good5$A_4\times C_2$ \( 1 - T - T^{2} + 19 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 9 T + 41 T^{2} + 125 T^{3} + 41 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 5 T + 25 T^{2} - 111 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 8 T + 63 T^{2} - 264 T^{3} + 63 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 4 T + 25 T^{2} - 88 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 41 T^{2} - 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 11 T + 111 T^{2} + 21 p T^{3} + 111 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 5 T + 43 T^{2} + 185 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 8 T + 67 T^{2} - 584 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 2 T + 59 T^{2} + 68 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 12 T + 149 T^{2} - 928 T^{3} + 149 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 4 T + 109 T^{2} - 312 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 5 T + 123 T^{2} + 573 T^{3} + 123 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 5 T + 141 T^{2} + 423 T^{3} + 141 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 22 T + 335 T^{2} - 3012 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 6 T + 185 T^{2} + 700 T^{3} + 185 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 18 T + 293 T^{2} + 2548 T^{3} + 293 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 13 T + 189 T^{2} - 1885 T^{3} + 189 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 13 T + 121 T^{2} - 591 T^{3} + 121 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 14 T + 211 T^{2} + 2436 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 23 T + 381 T^{2} - 47 p T^{3} + 381 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.54731845731829358869869916313, −7.40088290534798144864922821286, −7.10496244583181095748643657544, −6.77692884373167467615924820139, −6.46027811259837631275995955708, −6.33350151392703872128613239494, −6.21485427390476060566804959871, −5.81740366709497716364741588753, −5.72815471661042810775843274002, −5.68173206887183726973449170443, −5.00743096521024373364367139505, −4.97930912309709272831974312013, −4.64233172911181223659831621378, −3.90933235807965602446349059993, −3.83166109917097479146192851657, −3.71605848398281239141226939974, −3.51213503273680534968701904817, −3.35075013708881808024916840788, −2.95609531387861872304480538866, −2.49410971371330412200432732220, −2.45100061847091525661344848353, −1.98332225914900656051999627333, −1.29036641815871224992256545779, −1.02761441849840946925621366088, −0.52005572830403553416617388417, 0.52005572830403553416617388417, 1.02761441849840946925621366088, 1.29036641815871224992256545779, 1.98332225914900656051999627333, 2.45100061847091525661344848353, 2.49410971371330412200432732220, 2.95609531387861872304480538866, 3.35075013708881808024916840788, 3.51213503273680534968701904817, 3.71605848398281239141226939974, 3.83166109917097479146192851657, 3.90933235807965602446349059993, 4.64233172911181223659831621378, 4.97930912309709272831974312013, 5.00743096521024373364367139505, 5.68173206887183726973449170443, 5.72815471661042810775843274002, 5.81740366709497716364741588753, 6.21485427390476060566804959871, 6.33350151392703872128613239494, 6.46027811259837631275995955708, 6.77692884373167467615924820139, 7.10496244583181095748643657544, 7.40088290534798144864922821286, 7.54731845731829358869869916313

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