Properties

Label 6-2160e3-1.1-c3e3-0-7
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 + 8·7-s + 10·11-s + 48·13-s + 37·17-s − 29·19-s + 11·23-s + 150·25-s + 28·29-s − 41·31-s + 120·35-s + 230·37-s + 370·41-s + 130·43-s + 56·47-s − 209·49-s + 805·53-s + 150·55-s − 576·59-s − 257·61-s + 720·65-s + 14·67-s − 1.23e3·71-s − 398·73-s + 80·77-s + 321·79-s − 687·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.431·7-s + 0.274·11-s + 1.02·13-s + 0.527·17-s − 0.350·19-s + 0.0997·23-s + 6/5·25-s + 0.179·29-s − 0.237·31-s + 0.579·35-s + 1.02·37-s + 1.40·41-s + 0.461·43-s + 0.173·47-s − 0.609·49-s + 2.08·53-s + 0.367·55-s − 1.27·59-s − 0.539·61-s + 1.37·65-s + 0.0255·67-s − 2.06·71-s − 0.638·73-s + 0.118·77-s + 0.457·79-s − 0.908·83-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

\(L(2)\) \(\approx\) \(13.08184863\)
\(L(\frac12)\) \(\approx\) \(13.08184863\)
\(L(\frac{5}{2})\) 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$S_4\times C_2$ \( 1 - 8 T + 39 p T^{2} - 11320 T^{3} + 39 p^{4} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 10 T + 1129 T^{2} + 2980 p T^{3} + 1129 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 48 T + 3891 T^{2} - 92328 T^{3} + 3891 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 37 T + 14418 T^{2} - 347965 T^{3} + 14418 p^{3} T^{4} - 37 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 29 T + 17960 T^{2} + 420497 T^{3} + 17960 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 11 T + 30456 T^{2} - 222899 T^{3} + 30456 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 28 T + 55099 T^{2} - 1662544 T^{3} + 55099 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 41 T + 78964 T^{2} + 2734453 T^{3} + 78964 p^{3} T^{4} + 41 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 230 T + 114787 T^{2} - 19298596 T^{3} + 114787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 370 T + 214275 T^{2} - 48796108 T^{3} + 214275 p^{3} T^{4} - 370 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 130 T + 44673 T^{2} - 3936068 T^{3} + 44673 p^{3} T^{4} - 130 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 56 T + 115517 T^{2} - 4195536 T^{3} + 115517 p^{3} T^{4} - 56 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 805 T + 594702 T^{2} - 247415005 T^{3} + 594702 p^{3} T^{4} - 805 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 576 T + 3639 p T^{2} + 9306072 T^{3} + 3639 p^{4} T^{4} + 576 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 257 T + 202474 T^{2} + 153986701 T^{3} + 202474 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 14 T + 651149 T^{2} + 22117636 T^{3} + 651149 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 1238 T + 1448677 T^{2} + 895305068 T^{3} + 1448677 p^{3} T^{4} + 1238 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 398 T + 1211307 T^{2} + 310732420 T^{3} + 1211307 p^{3} T^{4} + 398 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 321 T + 774636 T^{2} - 459949637 T^{3} + 774636 p^{3} T^{4} - 321 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 687 T + 1037148 T^{2} + 349696983 T^{3} + 1037148 p^{3} T^{4} + 687 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 2358 T + 3793395 T^{2} - 3699365060 T^{3} + 3793395 p^{3} T^{4} - 2358 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 576 T + 2295459 T^{2} - 793944704 T^{3} + 2295459 p^{3} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.900709345842155104201924872688, −7.35988822784565074292177760372, −7.08781016933554089076732030156, −7.06946208982433766719888245251, −6.40353320112490616042641398705, −6.31546820908717618108096825165, −6.18797086675618530305834167098, −5.64643679931311375098650570035, −5.63471595729629378866781393814, −5.62811234306282118358385188633, −4.83406927950647403462889380298, −4.69883424859375951691219361431, −4.60738949418755871802913163234, −4.05420387706969139084496259359, −3.78712464588099908583269822776, −3.64594505422886909401462543489, −2.87447163526882839701804118320, −2.84300705714419785474799367710, −2.71282777438757283590115649406, −1.91863461292460923064782617960, −1.82454083217973335123660263679, −1.60026692718911216432123797919, −1.04705231529672536272627922277, −0.67727262051236517908123904869, −0.52804464620693375722439234959, 0.52804464620693375722439234959, 0.67727262051236517908123904869, 1.04705231529672536272627922277, 1.60026692718911216432123797919, 1.82454083217973335123660263679, 1.91863461292460923064782617960, 2.71282777438757283590115649406, 2.84300705714419785474799367710, 2.87447163526882839701804118320, 3.64594505422886909401462543489, 3.78712464588099908583269822776, 4.05420387706969139084496259359, 4.60738949418755871802913163234, 4.69883424859375951691219361431, 4.83406927950647403462889380298, 5.62811234306282118358385188633, 5.63471595729629378866781393814, 5.64643679931311375098650570035, 6.18797086675618530305834167098, 6.31546820908717618108096825165, 6.40353320112490616042641398705, 7.06946208982433766719888245251, 7.08781016933554089076732030156, 7.35988822784565074292177760372, 7.900709345842155104201924872688

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