Properties

Label 6-4560e3-1.1-c1e3-0-2
Degree $6$
Conductor $94818816000$
Sign $1$
Analytic cond. $48275.3$
Root an. cond. $6.03421$
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 + 2·7-s + 6·9-s − 4·11-s − 6·13-s − 9·15-s − 3·19-s + 6·21-s + 8·23-s + 6·25-s + 10·27-s + 10·29-s + 6·31-s − 12·33-s − 6·35-s − 6·37-s − 18·39-s + 4·41-s − 18·45-s + 16·47-s + 49-s + 22·53-s + 12·55-s − 9·57-s − 22·59-s + 2·61-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s − 1.20·11-s − 1.66·13-s − 2.32·15-s − 0.688·19-s + 1.30·21-s + 1.66·23-s + 6/5·25-s + 1.92·27-s + 1.85·29-s + 1.07·31-s − 2.08·33-s − 1.01·35-s − 0.986·37-s − 2.88·39-s + 0.624·41-s − 2.68·45-s + 2.33·47-s + 1/7·49-s + 3.02·53-s + 1.61·55-s − 1.19·57-s − 2.86·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(48275.3\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.980744677\)
\(L(\frac12)\) \(\approx\) \(6.980744677\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - 2 T + 3 T^{2} + 16 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 19 T^{2} + 96 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 6 T + 41 T^{2} + 152 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 11 T^{2} + 64 T^{3} + 11 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 65 T^{2} - 352 T^{3} + 65 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 10 T + 69 T^{2} - 312 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 6 T + 65 T^{2} - 236 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 6 T + 33 T^{2} + 56 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 4 T + 33 T^{2} + 72 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 39 T^{2} - 216 T^{3} + 39 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 16 T + 201 T^{2} - 1488 T^{3} + 201 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 22 T + 295 T^{2} - 2508 T^{3} + 295 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 22 T + 217 T^{2} + 1620 T^{3} + 217 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 87 T^{2} - 404 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 4 T + 105 T^{2} + 664 T^{3} + 105 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 29 T^{2} + 464 T^{3} + 29 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 119 T^{2} - 972 T^{3} + 119 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 10 T + 3 p T^{2} - 1516 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 2 T + 149 T^{2} + 4 T^{3} + 149 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 24 T + 369 T^{2} - 3848 T^{3} + 369 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 6 T + 213 T^{2} + 8 p T^{3} + 213 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.60800863840146645619356806621, −7.18911565528530066651119914657, −7.12785741386052720070298924439, −6.99859155621159722215605466215, −6.26697278607794530796168634103, −6.23820889168846001023862730722, −6.17474574443810042462218917804, −5.36487259819267208050322338016, −5.13213202654657089531764929083, −5.12670754800388322979682799453, −4.67686511391186482739441751976, −4.56101495918939957952013174343, −4.48447841819177321933274404445, −4.01925288869820480948380216860, −3.69006393834102253877941099068, −3.49469238893022622688385567644, −3.02793993035403799028998483863, −2.98219200991965002596393503180, −2.57564133386498495443267327782, −2.27699329572006245533550470561, −2.25004817037691356095910465772, −1.81029201872547652062376562438, −1.04619841104293910151690863677, −0.812554832927138809294036654307, −0.50618660957208422987682212753, 0.50618660957208422987682212753, 0.812554832927138809294036654307, 1.04619841104293910151690863677, 1.81029201872547652062376562438, 2.25004817037691356095910465772, 2.27699329572006245533550470561, 2.57564133386498495443267327782, 2.98219200991965002596393503180, 3.02793993035403799028998483863, 3.49469238893022622688385567644, 3.69006393834102253877941099068, 4.01925288869820480948380216860, 4.48447841819177321933274404445, 4.56101495918939957952013174343, 4.67686511391186482739441751976, 5.12670754800388322979682799453, 5.13213202654657089531764929083, 5.36487259819267208050322338016, 6.17474574443810042462218917804, 6.23820889168846001023862730722, 6.26697278607794530796168634103, 6.99859155621159722215605466215, 7.12785741386052720070298924439, 7.18911565528530066651119914657, 7.60800863840146645619356806621

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