Properties

Label 6-2160e3-1.1-c3e3-0-1
Degree $6$
Conductor $10077696000$
Sign $1$
Analytic cond. $2.06994\times 10^{6}$
Root an. cond. $11.2891$
Motivic weight $3$
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  + 15·5-s − 44·7-s − 38·11-s + 28·13-s − 19·17-s − 187·19-s + 81·23-s + 150·25-s + 160·29-s − 227·31-s − 660·35-s + 78·37-s − 338·41-s − 22·43-s + 472·47-s + 355·49-s + 521·53-s − 570·55-s − 140·59-s + 595·61-s + 420·65-s − 878·67-s + 602·71-s + 1.29e3·73-s + 1.67e3·77-s − 629·79-s + 1.28e3·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 2.37·7-s − 1.04·11-s + 0.597·13-s − 0.271·17-s − 2.25·19-s + 0.734·23-s + 6/5·25-s + 1.02·29-s − 1.31·31-s − 3.18·35-s + 0.346·37-s − 1.28·41-s − 0.0780·43-s + 1.46·47-s + 1.03·49-s + 1.35·53-s − 1.39·55-s − 0.308·59-s + 1.24·61-s + 0.801·65-s − 1.60·67-s + 1.00·71-s + 2.07·73-s + 2.47·77-s − 0.895·79-s + 1.70·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(2.06994\times 10^{6}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.708631814\)
\(L(\frac12)\) \(\approx\) \(2.708631814\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 44 T + 1581 T^{2} + 31984 T^{3} + 1581 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 38 T + 1381 T^{2} + 17876 T^{3} + 1381 p^{3} T^{4} + 38 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 28 T + 155 p T^{2} - 22912 T^{3} + 155 p^{4} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 19 T + 3262 T^{2} - 367193 T^{3} + 3262 p^{3} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 187 T + 24164 T^{2} + 2039395 T^{3} + 24164 p^{3} T^{4} + 187 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 81 T + 13200 T^{2} + 72927 T^{3} + 13200 p^{3} T^{4} - 81 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 160 T + 25399 T^{2} + 88280 T^{3} + 25399 p^{3} T^{4} - 160 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 227 T + 71400 T^{2} + 13771435 T^{3} + 71400 p^{3} T^{4} + 227 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 78 T + 52035 T^{2} + 5735212 T^{3} + 52035 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 338 T + 163951 T^{2} + 34473956 T^{3} + 163951 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 22 T + 78605 T^{2} - 14966252 T^{3} + 78605 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 472 T + 365677 T^{2} - 97725712 T^{3} + 365677 p^{3} T^{4} - 472 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 521 T + 508018 T^{2} - 154190045 T^{3} + 508018 p^{3} T^{4} - 521 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 140 T + 449689 T^{2} + 23374640 T^{3} + 449689 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 878 T + 978237 T^{2} + 516844828 T^{3} + 978237 p^{3} T^{4} + 878 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 602 T + 490081 T^{2} - 150373964 T^{3} + 490081 p^{3} T^{4} - 602 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1294 T + 1071143 T^{2} - 602684716 T^{3} + 1071143 p^{3} T^{4} - 1294 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 629 T + 1576176 T^{2} + 622253365 T^{3} + 1576176 p^{3} T^{4} + 629 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 1287 T + 1349484 T^{2} - 1125375327 T^{3} + 1349484 p^{3} T^{4} - 1287 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 2154 T + 3172479 T^{2} - 3111332052 T^{3} + 3172479 p^{3} T^{4} - 2154 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1392 T + 3281955 T^{2} - 2604477152 T^{3} + 3281955 p^{3} T^{4} - 1392 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.78334205277395964477329231318, −7.31258320542622850644462473984, −6.89577503125955871878171796098, −6.85863715165637824503871969014, −6.58320588219710314558331443669, −6.33457049583951521481502345100, −6.29301487779999469931204789154, −5.84973013745685646256138464053, −5.58591978131943322965240682008, −5.50989989442889908482387859375, −4.89600460652895144887004125287, −4.74678769964716034633067409205, −4.62624697465583458001132783024, −3.79504570544272951366196800095, −3.72109248109049723432458723942, −3.66917888944209187472647228521, −3.05244187901225877553309956917, −2.77308847252182522952751599157, −2.62736930094076322291882455720, −2.14774120724727050357021016316, −1.94534482769716810466713103830, −1.62265048657767930799331853594, −0.843022139899256291757632867045, −0.56695036829938167538167713089, −0.30719045186023381568600411022, 0.30719045186023381568600411022, 0.56695036829938167538167713089, 0.843022139899256291757632867045, 1.62265048657767930799331853594, 1.94534482769716810466713103830, 2.14774120724727050357021016316, 2.62736930094076322291882455720, 2.77308847252182522952751599157, 3.05244187901225877553309956917, 3.66917888944209187472647228521, 3.72109248109049723432458723942, 3.79504570544272951366196800095, 4.62624697465583458001132783024, 4.74678769964716034633067409205, 4.89600460652895144887004125287, 5.50989989442889908482387859375, 5.58591978131943322965240682008, 5.84973013745685646256138464053, 6.29301487779999469931204789154, 6.33457049583951521481502345100, 6.58320588219710314558331443669, 6.85863715165637824503871969014, 6.89577503125955871878171796098, 7.31258320542622850644462473984, 7.78334205277395964477329231318

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