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Properties

Label 12-570e6-1.1-c1e6-0-7
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  + 9·7-s − 8-s − 9·11-s + 9·13-s − 9·17-s − 9·19-s + 3·23-s − 27-s − 9·29-s + 18·31-s − 6·37-s − 6·41-s + 21·43-s + 45·49-s + 12·53-s − 9·56-s + 9·59-s + 3·61-s + 36·67-s − 36·71-s + 21·73-s − 81·77-s − 9·79-s − 3·83-s + 9·88-s + 3·89-s + 81·91-s + ⋯
L(s)  = 1  + 3.40·7-s − 0.353·8-s − 2.71·11-s + 2.49·13-s − 2.18·17-s − 2.06·19-s + 0.625·23-s − 0.192·27-s − 1.67·29-s + 3.23·31-s − 0.986·37-s − 0.937·41-s + 3.20·43-s + 45/7·49-s + 1.64·53-s − 1.20·56-s + 1.17·59-s + 0.384·61-s + 4.39·67-s − 4.27·71-s + 2.45·73-s − 9.23·77-s − 1.01·79-s − 0.329·83-s + 0.959·88-s + 0.317·89-s + 8.49·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.047768351\)
\(L(\frac12)\) \(\approx\) \(5.047768351\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
3 \( 1 + T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( 1 + 9 T - 179 T^{3} + 9 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 - 9 T + 36 T^{2} - 115 T^{3} + 405 T^{4} - 1296 T^{5} + 3567 T^{6} - 1296 p T^{7} + 405 p^{2} T^{8} - 115 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 9 T + 24 T^{2} + 83 T^{3} + 687 T^{4} + 2058 T^{5} + 3347 T^{6} + 2058 p T^{7} + 687 p^{2} T^{8} + 83 p^{3} T^{9} + 24 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 72 T^{3} - 153 T^{4} + 2493 T^{5} - 12295 T^{6} + 2493 p T^{7} - 153 p^{2} T^{8} - 72 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 54 T^{2} + 310 T^{3} + 1593 T^{4} + 6651 T^{5} + 27629 T^{6} + 6651 p T^{7} + 1593 p^{2} T^{8} + 310 p^{3} T^{9} + 54 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T + 9 T^{2} + 45 T^{3} - 18 T^{4} - 2514 T^{5} + 19441 T^{6} - 2514 p T^{7} - 18 p^{2} T^{8} + 45 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 45 T^{2} + 167 T^{3} - 468 T^{4} - 9612 T^{5} - 59275 T^{6} - 9612 p T^{7} - 468 p^{2} T^{8} + 167 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 18 T + 180 T^{2} - 938 T^{3} + 1386 T^{4} + 24318 T^{5} - 210669 T^{6} + 24318 p T^{7} + 1386 p^{2} T^{8} - 938 p^{3} T^{9} + 180 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 57 T^{2} + 59 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 6 T + 30 T^{2} + 569 T^{3} + 2289 T^{4} + 13131 T^{5} + 193757 T^{6} + 13131 p T^{7} + 2289 p^{2} T^{8} + 569 p^{3} T^{9} + 30 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 21 T + 132 T^{2} + 226 T^{3} - 4536 T^{4} - 26217 T^{5} + 468693 T^{6} - 26217 p T^{7} - 4536 p^{2} T^{8} + 226 p^{3} T^{9} + 132 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 36 T^{2} + 333 T^{3} + 63 p T^{4} + 16155 T^{5} + 123409 T^{6} + 16155 p T^{7} + 63 p^{3} T^{8} + 333 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 12 T + 99 T^{2} - 531 T^{3} + 837 T^{4} - 33 p T^{5} - 154 p T^{6} - 33 p^{2} T^{7} + 837 p^{2} T^{8} - 531 p^{3} T^{9} + 99 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T + 36 T^{2} - 230 T^{3} - 504 T^{4} + 585 p T^{5} - 365563 T^{6} + 585 p^{2} T^{7} - 504 p^{2} T^{8} - 230 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T + 12 T^{2} + 266 T^{3} - 1053 T^{4} - 23301 T^{5} + 301965 T^{6} - 23301 p T^{7} - 1053 p^{2} T^{8} + 266 p^{3} T^{9} + 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 36 T + 738 T^{2} - 10919 T^{3} + 129483 T^{4} - 1299717 T^{5} + 11339841 T^{6} - 1299717 p T^{7} + 129483 p^{2} T^{8} - 10919 p^{3} T^{9} + 738 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 36 T + 630 T^{2} + 7515 T^{3} + 71163 T^{4} + 580761 T^{5} + 4675537 T^{6} + 580761 p T^{7} + 71163 p^{2} T^{8} + 7515 p^{3} T^{9} + 630 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 21 T + 264 T^{2} - 1984 T^{3} + 11691 T^{4} - 70443 T^{5} + 534417 T^{6} - 70443 p T^{7} + 11691 p^{2} T^{8} - 1984 p^{3} T^{9} + 264 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 9 T + 81 T^{2} + 1465 T^{3} + 11898 T^{4} + 87246 T^{5} + 1046577 T^{6} + 87246 p T^{7} + 11898 p^{2} T^{8} + 1465 p^{3} T^{9} + 81 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 3 T - 150 T^{2} - 373 T^{3} + 10809 T^{4} + 9060 T^{5} - 836125 T^{6} + 9060 p T^{7} + 10809 p^{2} T^{8} - 373 p^{3} T^{9} - 150 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T - 108 T^{2} + 1224 T^{3} - 8190 T^{4} - 86853 T^{5} + 1870147 T^{6} - 86853 p T^{7} - 8190 p^{2} T^{8} + 1224 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 1694 T^{3} + 1956963 T^{6} - 1694 p^{3} T^{9} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.70775124689053019189277027109, −5.51625255496682729149524508781, −5.43608379323517801041077236541, −5.28273285184406116766726755047, −5.13997123247020908841861275462, −4.75309695404728942923969092000, −4.64753316825832257944332092087, −4.62238544235471097394706443349, −4.52685340048696072569636856266, −4.20144233494968550037507391814, −3.87553885475496998705710391757, −3.85098607040474288844192970026, −3.84304940056845913339519240997, −3.57277847224967723669613116260, −2.99932161110703699024559590419, −2.75544648098091635607753378770, −2.60750199481775273155705840305, −2.46902093533486003756647698944, −2.24979102130868326308102986062, −2.15078941611350843788293660605, −1.73323821703886872615014676849, −1.60340675070204504926089373819, −1.26788868023567438236746411225, −0.68739039535476345136347973907, −0.58270084889359487874146082038, 0.58270084889359487874146082038, 0.68739039535476345136347973907, 1.26788868023567438236746411225, 1.60340675070204504926089373819, 1.73323821703886872615014676849, 2.15078941611350843788293660605, 2.24979102130868326308102986062, 2.46902093533486003756647698944, 2.60750199481775273155705840305, 2.75544648098091635607753378770, 2.99932161110703699024559590419, 3.57277847224967723669613116260, 3.84304940056845913339519240997, 3.85098607040474288844192970026, 3.87553885475496998705710391757, 4.20144233494968550037507391814, 4.52685340048696072569636856266, 4.62238544235471097394706443349, 4.64753316825832257944332092087, 4.75309695404728942923969092000, 5.13997123247020908841861275462, 5.28273285184406116766726755047, 5.43608379323517801041077236541, 5.51625255496682729149524508781, 5.70775124689053019189277027109

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

Plot not available for L-functions of degree greater than 10.