Properties

Label 6-2160e3-1.1-c3e3-0-15
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 $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·5-s − 9·7-s − 18·11-s − 21·13-s + 84·17-s − 21·19-s − 48·23-s + 150·25-s − 36·29-s − 324·31-s − 135·35-s + 33·37-s + 114·41-s − 282·43-s − 282·47-s − 360·49-s + 222·53-s − 270·55-s + 276·59-s + 303·61-s − 315·65-s − 1.03e3·67-s − 510·71-s + 447·73-s + 162·77-s − 777·79-s − 78·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.485·7-s − 0.493·11-s − 0.448·13-s + 1.19·17-s − 0.253·19-s − 0.435·23-s + 6/5·25-s − 0.230·29-s − 1.87·31-s − 0.651·35-s + 0.146·37-s + 0.434·41-s − 1.00·43-s − 0.875·47-s − 1.04·49-s + 0.575·53-s − 0.661·55-s + 0.609·59-s + 0.635·61-s − 0.601·65-s − 1.88·67-s − 0.852·71-s + 0.716·73-s + 0.239·77-s − 1.10·79-s − 0.103·83-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: induced by $\chi_{2160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

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$S_4\times C_2$ \( 1 + 9 T + 9 p^{2} T^{2} + 650 T^{3} + 9 p^{5} T^{4} + 9 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 18 T + 912 T^{2} - 28510 T^{3} + 912 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 21 T + 2910 T^{2} + 4721 p T^{3} + 2910 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 84 T + 14706 T^{2} - 738652 T^{3} + 14706 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 21 T + 15189 T^{2} + 149614 T^{3} + 15189 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 48 T + 36654 T^{2} + 1165994 T^{3} + 36654 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 36 T + 31512 T^{2} - 1278578 T^{3} + 31512 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 324 T + 70560 T^{2} + 10091392 T^{3} + 70560 p^{3} T^{4} + 324 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 33 T + 107211 T^{2} - 491142 T^{3} + 107211 p^{3} T^{4} - 33 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 114 T + 125619 T^{2} - 13866756 T^{3} + 125619 p^{3} T^{4} - 114 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 282 T + 220578 T^{2} + 43728712 T^{3} + 220578 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 6 p T + 152826 T^{2} + 69794468 T^{3} + 152826 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 222 T + 282615 T^{2} - 72010284 T^{3} + 282615 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \)
59$C_2$ \( ( 1 - 92 T + p^{3} T^{2} )^{3} \)
61$S_4\times C_2$ \( 1 - 303 T + 620067 T^{2} - 124220266 T^{3} + 620067 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 1035 T + 63873 T^{2} - 227096926 T^{3} + 63873 p^{3} T^{4} + 1035 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 510 T + 1098285 T^{2} + 353629860 T^{3} + 1098285 p^{3} T^{4} + 510 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 447 T + 657111 T^{2} - 269206770 T^{3} + 657111 p^{3} T^{4} - 447 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 777 T + 14268 p T^{2} + 780182781 T^{3} + 14268 p^{4} T^{4} + 777 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 78 T + 882717 T^{2} + 176745340 T^{3} + 882717 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 324 T + 1606299 T^{2} + 549274824 T^{3} + 1606299 p^{3} T^{4} + 324 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1191 T + 2555007 T^{2} - 2142250738 T^{3} + 2555007 p^{3} T^{4} - 1191 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.031045062360317698133057044264, −7.74344585912297469793807880142, −7.65493191160554565003249858373, −7.31858737809018870030372597318, −6.97557442620747893821136293447, −6.72296317108167638162863543011, −6.53798021857457575348458508033, −6.12629412290516959916428941785, −6.07736387890552511899348628238, −5.58566976342226000246787325249, −5.30887165113565709222079893066, −5.29094342185355020311232801911, −5.15965170847770020667507249676, −4.47182675511791707609890332874, −4.32838531750722442892045758828, −3.97956198606435513412392390498, −3.51203911652305220053848828474, −3.23069917940648544562306349341, −3.16947442896766078597779118816, −2.47304972560274849281364639734, −2.32885282283661426561726151817, −2.21502325030531071594558919439, −1.42591659551756577157957946280, −1.25664347571108240023114863116, −1.23102024427839059641217786684, 0, 0, 0, 1.23102024427839059641217786684, 1.25664347571108240023114863116, 1.42591659551756577157957946280, 2.21502325030531071594558919439, 2.32885282283661426561726151817, 2.47304972560274849281364639734, 3.16947442896766078597779118816, 3.23069917940648544562306349341, 3.51203911652305220053848828474, 3.97956198606435513412392390498, 4.32838531750722442892045758828, 4.47182675511791707609890332874, 5.15965170847770020667507249676, 5.29094342185355020311232801911, 5.30887165113565709222079893066, 5.58566976342226000246787325249, 6.07736387890552511899348628238, 6.12629412290516959916428941785, 6.53798021857457575348458508033, 6.72296317108167638162863543011, 6.97557442620747893821136293447, 7.31858737809018870030372597318, 7.65493191160554565003249858373, 7.74344585912297469793807880142, 8.031045062360317698133057044264

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