Properties

Degree $12$
Conductor $7.647\times 10^{20}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 11-s + 8·13-s + 4·17-s + 3·19-s − 7·23-s + 9·25-s + 5·29-s + 40·31-s + 2·35-s + 3·37-s + 6·43-s + 18·47-s + 8·49-s − 15·53-s + 55-s + 28·59-s − 16·61-s − 8·65-s + 2·67-s + 14·71-s + 19·73-s + 2·77-s + 10·79-s + 2·83-s − 4·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.301·11-s + 2.21·13-s + 0.970·17-s + 0.688·19-s − 1.45·23-s + 9/5·25-s + 0.928·29-s + 7.18·31-s + 0.338·35-s + 0.493·37-s + 0.914·43-s + 2.62·47-s + 8/7·49-s − 2.06·53-s + 0.134·55-s + 3.64·59-s − 2.04·61-s − 0.992·65-s + 0.244·67-s + 1.66·71-s + 2.22·73-s + 0.227·77-s + 1.12·79-s + 0.219·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(26.18537853\)
\(L(\frac12)\) \(\approx\) \(26.18537853\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 2 T - 4 T^{2} - 31 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 52 p T^{7} + 23 p^{2} T^{8} - 17 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 26 T^{2} - 23 T^{3} + 37 p T^{4} + 202 T^{5} - 4853 T^{6} + 202 p T^{7} + 37 p^{3} T^{8} - 23 p^{3} T^{9} - 26 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 23 T^{2} + 4 p T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 220 p T^{7} + 410 p^{2} T^{8} + 4 p^{4} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 54 p T^{7} - 153 p^{2} T^{8} + 67 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 3946 p T^{7} + 2423 p^{2} T^{8} - 83 p^{3} T^{9} - 32 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 20 T + 214 T^{2} - 1441 T^{3} + 214 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
41 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 6 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 60360 p T^{7} + 13635 p^{2} T^{8} + 33 p^{3} T^{9} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 216 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - T + 89 T^{2} + 77 T^{3} + 89 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 15 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 - 5 T + 163 T^{2} - 469 T^{3} + 163 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 13564 p T^{7} + 18788 p^{2} T^{8} - 2 p^{3} T^{9} - 182 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 73314 p T^{7} + 16101 p^{2} T^{8} + 1197 p^{3} T^{9} - 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.69237006654212370339875906609, −4.14504608451351355454099928728, −4.12635961907907185290580165936, −3.98038719368171305050567111948, −3.96796592622568655131583711415, −3.89951364282526885830926863199, −3.75974650312896350434505666452, −3.31371380634973596045311532899, −3.30555379565931250640606262639, −3.05615273573174483146746531650, −3.04395319062033234423858849295, −2.92538060991097997387531344602, −2.87857819222402069776386188015, −2.44395589015714153884188980307, −2.29756603743776511745785500798, −2.27647777965153986903325256728, −2.18169736033481201028030155186, −1.78792542907884659169624944994, −1.60961972966744631770239461711, −1.14540992544256778999823923179, −1.00000448501820043552246953722, −0.818978873605584258411545165474, −0.798908989504580595444235234905, −0.65465749046276465582053241951, −0.63816550227816665519455639918, 0.63816550227816665519455639918, 0.65465749046276465582053241951, 0.798908989504580595444235234905, 0.818978873605584258411545165474, 1.00000448501820043552246953722, 1.14540992544256778999823923179, 1.60961972966744631770239461711, 1.78792542907884659169624944994, 2.18169736033481201028030155186, 2.27647777965153986903325256728, 2.29756603743776511745785500798, 2.44395589015714153884188980307, 2.87857819222402069776386188015, 2.92538060991097997387531344602, 3.04395319062033234423858849295, 3.05615273573174483146746531650, 3.30555379565931250640606262639, 3.31371380634973596045311532899, 3.75974650312896350434505666452, 3.89951364282526885830926863199, 3.96796592622568655131583711415, 3.98038719368171305050567111948, 4.12635961907907185290580165936, 4.14504608451351355454099928728, 4.69237006654212370339875906609

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

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