Properties

Label 12-1232e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.497\times 10^{18}$
Sign $1$
Analytic cond. $906414.$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 2·5-s + 15·9-s + 3·11-s − 6·13-s − 10·15-s + 11·17-s + 7·19-s − 6·23-s − 38·27-s − 6·29-s + 19·31-s − 15·33-s − 12·37-s + 30·39-s − 34·41-s − 24·43-s + 30·45-s − 17·47-s − 55·51-s + 3·53-s + 6·55-s − 35·57-s − 12·59-s − 12·65-s + 30·69-s + 6·71-s + ⋯
L(s)  = 1  − 2.88·3-s + 0.894·5-s + 5·9-s + 0.904·11-s − 1.66·13-s − 2.58·15-s + 2.66·17-s + 1.60·19-s − 1.25·23-s − 7.31·27-s − 1.11·29-s + 3.41·31-s − 2.61·33-s − 1.97·37-s + 4.80·39-s − 5.30·41-s − 3.65·43-s + 4.47·45-s − 2.47·47-s − 7.70·51-s + 0.412·53-s + 0.809·55-s − 4.63·57-s − 1.56·59-s − 1.48·65-s + 3.61·69-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(906414.\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.033565560\)
\(L(\frac12)\) \(\approx\) \(1.033565560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - p T^{3} + p^{3} T^{6} \)
11 \( ( 1 - T + T^{2} )^{3} \)
good3 \( 1 + 5 T + 10 T^{2} + 13 T^{3} + 25 T^{4} + 20 p T^{5} + 115 T^{6} + 20 p^{2} T^{7} + 25 p^{2} T^{8} + 13 p^{3} T^{9} + 10 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 4 T^{2} - 18 T^{3} + 18 T^{4} - 2 p^{2} T^{5} + 231 T^{6} - 2 p^{3} T^{7} + 18 p^{2} T^{8} - 18 p^{3} T^{9} + 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 35 T^{2} + 79 T^{3} + 35 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 11 T + 46 T^{2} - 135 T^{3} + 555 T^{4} - 1004 T^{5} - 1743 T^{6} - 1004 p T^{7} + 555 p^{2} T^{8} - 135 p^{3} T^{9} + 46 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 7 T - 22 T^{2} + 49 T^{3} + 1781 T^{4} - 3024 T^{5} - 25221 T^{6} - 3024 p T^{7} + 1781 p^{2} T^{8} + 49 p^{3} T^{9} - 22 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T - 38 T^{2} - 110 T^{3} + 2112 T^{4} + 3248 T^{5} - 43929 T^{6} + 3248 p T^{7} + 2112 p^{2} T^{8} - 110 p^{3} T^{9} - 38 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 3 T + 41 T^{2} + 175 T^{3} + 41 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 19 T + 150 T^{2} - 1175 T^{3} + 11057 T^{4} - 68688 T^{5} + 340399 T^{6} - 68688 p T^{7} + 11057 p^{2} T^{8} - 1175 p^{3} T^{9} + 150 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 12 T + 13 T^{2} - 220 T^{3} + 1562 T^{4} + 17476 T^{5} + 72029 T^{6} + 17476 p T^{7} + 1562 p^{2} T^{8} - 220 p^{3} T^{9} + 13 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 17 T + 175 T^{2} + 1227 T^{3} + 175 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 12 T + 128 T^{2} + 949 T^{3} + 128 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 17 T + 166 T^{2} + 741 T^{3} - 2127 T^{4} - 75976 T^{5} - 686193 T^{6} - 75976 p T^{7} - 2127 p^{2} T^{8} + 741 p^{3} T^{9} + 166 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T - 104 T^{2} + 103 T^{3} + 6117 T^{4} + 2534 T^{5} - 355443 T^{6} + 2534 p T^{7} + 6117 p^{2} T^{8} + 103 p^{3} T^{9} - 104 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 12 T + 66 T^{2} + 146 T^{3} - 3258 T^{4} - 37866 T^{5} - 293437 T^{6} - 37866 p T^{7} - 3258 p^{2} T^{8} + 146 p^{3} T^{9} + 66 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 99 T^{2} + 112 T^{3} + 3762 T^{4} - 5544 T^{5} - 140763 T^{6} - 5544 p T^{7} + 3762 p^{2} T^{8} + 112 p^{3} T^{9} - 99 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 - 117 T^{2} + 112 T^{3} + 5850 T^{4} - 6552 T^{5} - 312501 T^{6} - 6552 p T^{7} + 5850 p^{2} T^{8} + 112 p^{3} T^{9} - 117 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 3 T + 83 T^{2} - 385 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 26 T + 290 T^{2} - 2218 T^{3} + 19184 T^{4} - 182172 T^{5} + 1579203 T^{6} - 182172 p T^{7} + 19184 p^{2} T^{8} - 2218 p^{3} T^{9} + 290 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 23 T + 118 T^{2} - 1319 T^{3} + 42911 T^{4} - 321506 T^{5} + 880319 T^{6} - 321506 p T^{7} + 42911 p^{2} T^{8} - 1319 p^{3} T^{9} + 118 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 9 T + 143 T^{2} - 821 T^{3} + 143 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 13 T - 138 T^{2} - 695 T^{3} + 36467 T^{4} + 111876 T^{5} - 2827223 T^{6} + 111876 p T^{7} + 36467 p^{2} T^{8} - 695 p^{3} T^{9} - 138 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - T + 107 T^{2} + 717 T^{3} + 107 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−5.07638070786186324655293127448, −5.06383763065036998344235460222, −4.96483364546031924879308255395, −4.96444045170169075151277012424, −4.56171677559984546014286443575, −4.53090578533709941475528276917, −4.42889990417116871192606825166, −3.82023982657698934813280963928, −3.69289426852122449675945534058, −3.62243767893325953301294037161, −3.58914835000238707346930926584, −3.39340330516321040791623290748, −3.34128823508960586785468011977, −2.90967689997981171227835206283, −2.78323748144451751770551985283, −2.60548399166133743172373616288, −1.97606491079758725697079694520, −1.95710828961918059767454793431, −1.70857201858715069688004473099, −1.61489880615058853853955354001, −1.60936787587088560422452538481, −1.18141492703124369138378666892, −0.861906003136127119256232684271, −0.43074756290414163619544551685, −0.30500631236416798870406962112, 0.30500631236416798870406962112, 0.43074756290414163619544551685, 0.861906003136127119256232684271, 1.18141492703124369138378666892, 1.60936787587088560422452538481, 1.61489880615058853853955354001, 1.70857201858715069688004473099, 1.95710828961918059767454793431, 1.97606491079758725697079694520, 2.60548399166133743172373616288, 2.78323748144451751770551985283, 2.90967689997981171227835206283, 3.34128823508960586785468011977, 3.39340330516321040791623290748, 3.58914835000238707346930926584, 3.62243767893325953301294037161, 3.69289426852122449675945534058, 3.82023982657698934813280963928, 4.42889990417116871192606825166, 4.53090578533709941475528276917, 4.56171677559984546014286443575, 4.96444045170169075151277012424, 4.96483364546031924879308255395, 5.06383763065036998344235460222, 5.07638070786186324655293127448

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

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