Properties

Label 6-1080e3-1.1-c3e3-0-10
Degree $6$
Conductor $1259712000$
Sign $-1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·5-s − 24·7-s + 6·11-s + 48·13-s − 27·17-s − 195·19-s − 27·23-s + 150·25-s + 60·29-s − 279·31-s − 360·35-s − 138·37-s + 66·41-s + 222·43-s − 264·47-s − 345·49-s − 507·53-s + 90·55-s − 960·59-s + 543·61-s + 720·65-s − 1.08e3·67-s − 1.81e3·71-s + 1.36e3·73-s − 144·77-s − 129·79-s − 1.56e3·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.29·7-s + 0.164·11-s + 1.02·13-s − 0.385·17-s − 2.35·19-s − 0.244·23-s + 6/5·25-s + 0.384·29-s − 1.61·31-s − 1.73·35-s − 0.613·37-s + 0.251·41-s + 0.787·43-s − 0.819·47-s − 1.00·49-s − 1.31·53-s + 0.220·55-s − 2.11·59-s + 1.13·61-s + 1.37·65-s − 1.98·67-s − 3.03·71-s + 2.18·73-s − 0.213·77-s − 0.183·79-s − 2.07·83-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$S_4\times C_2$ \( 1 + 24 T + 921 T^{2} + 12904 T^{3} + 921 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 6 T + 513 T^{2} - 60620 T^{3} + 513 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 48 T + 3435 T^{2} - 202552 T^{3} + 3435 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 27 T - 1158 T^{2} - 163141 T^{3} - 1158 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 195 T + 20832 T^{2} + 1720919 T^{3} + 20832 p^{3} T^{4} + 195 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 27 T - 168 T^{2} - 1130141 T^{3} - 168 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 60 T + 68067 T^{2} - 2620544 T^{3} + 68067 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 9 p T + 111132 T^{2} + 17041763 T^{3} + 111132 p^{3} T^{4} + 9 p^{7} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 138 T + 133923 T^{2} + 14102268 T^{3} + 133923 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 66 T + 63627 T^{2} + 6726804 T^{3} + 63627 p^{3} T^{4} - 66 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 222 T + 109929 T^{2} - 9784924 T^{3} + 109929 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 264 T + 124605 T^{2} + 55592752 T^{3} + 124605 p^{3} T^{4} + 264 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 507 T + 524166 T^{2} + 154386795 T^{3} + 524166 p^{3} T^{4} + 507 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 960 T + 843525 T^{2} + 409376120 T^{3} + 843525 p^{3} T^{4} + 960 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 543 T + 239946 T^{2} - 10210291 T^{3} + 239946 p^{3} T^{4} - 543 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 1086 T + 604461 T^{2} + 230866972 T^{3} + 604461 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 1818 T + 2146461 T^{2} + 1508229108 T^{3} + 2146461 p^{3} T^{4} + 1818 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1362 T + 1530147 T^{2} - 1054519260 T^{3} + 1530147 p^{3} T^{4} - 1362 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 129 T + 1152804 T^{2} + 62398365 T^{3} + 1152804 p^{3} T^{4} + 129 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 1569 T + 2366268 T^{2} + 1835088857 T^{3} + 2366268 p^{3} T^{4} + 1569 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 1770 T + 2057019 T^{2} + 1828728060 T^{3} + 2057019 p^{3} T^{4} + 1770 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 336 T + 1258275 T^{2} + 1117216928 T^{3} + 1258275 p^{3} T^{4} + 336 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.855298369238887270457962807610, −8.616413313791928044506622646051, −8.368788313502090377075880932202, −8.179126238446291767136661426644, −7.58631022200336632697029538722, −7.43534484928982465803650947384, −6.98675267659724483299472873984, −6.65149903783077632211193868746, −6.53587194462843666892800843929, −6.28511906109972511700367694204, −5.92762036368674000117397011264, −5.84508362651120888937787002222, −5.61547660224623181161128991606, −4.92227812153774725985181985803, −4.69339257953013455333637222275, −4.55855409487396979600220086853, −3.82862202276236538551606746883, −3.77850115524339003462973561477, −3.50448822698546988410254874957, −2.77923368776646260878030307642, −2.64059311950255450089875521625, −2.45817011220563662751963284146, −1.57940059891343308903227160333, −1.44941908736202297501412134645, −1.41637318733592999584625670901, 0, 0, 0, 1.41637318733592999584625670901, 1.44941908736202297501412134645, 1.57940059891343308903227160333, 2.45817011220563662751963284146, 2.64059311950255450089875521625, 2.77923368776646260878030307642, 3.50448822698546988410254874957, 3.77850115524339003462973561477, 3.82862202276236538551606746883, 4.55855409487396979600220086853, 4.69339257953013455333637222275, 4.92227812153774725985181985803, 5.61547660224623181161128991606, 5.84508362651120888937787002222, 5.92762036368674000117397011264, 6.28511906109972511700367694204, 6.53587194462843666892800843929, 6.65149903783077632211193868746, 6.98675267659724483299472873984, 7.43534484928982465803650947384, 7.58631022200336632697029538722, 8.179126238446291767136661426644, 8.368788313502090377075880932202, 8.616413313791928044506622646051, 8.855298369238887270457962807610

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