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Properties

Label 12-42e12-1.1-c1e6-0-7
Degree $12$
Conductor $3.013\times 10^{19}$
Sign $1$
Analytic cond. $7.81005\times 10^{6}$
Root an. cond. $3.75308$
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  + 4·3-s − 5-s + 6·9-s − 2·11-s − 3·13-s − 4·15-s − 2·17-s − 3·19-s − 14·23-s + 11·25-s + 5·27-s − 29-s − 6·31-s − 8·33-s + 3·37-s − 12·39-s − 3·43-s − 6·45-s + 42·47-s − 8·51-s − 6·53-s + 2·55-s − 12·57-s + 62·59-s + 12·61-s + 3·65-s + 12·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.447·5-s + 2·9-s − 0.603·11-s − 0.832·13-s − 1.03·15-s − 0.485·17-s − 0.688·19-s − 2.91·23-s + 11/5·25-s + 0.962·27-s − 0.185·29-s − 1.07·31-s − 1.39·33-s + 0.493·37-s − 1.92·39-s − 0.457·43-s − 0.894·45-s + 6.12·47-s − 1.12·51-s − 0.824·53-s + 0.269·55-s − 1.58·57-s + 8.07·59-s + 1.53·61-s + 0.372·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(7.81005\times 10^{6}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.39315360\)
\(L(\frac12)\) \(\approx\) \(14.39315360\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + 10 T^{2} - 7 p T^{3} + 10 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good5 \( 1 + T - 2 p T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 14 p T^{7} + 57 p^{2} T^{8} - 7 p^{3} T^{9} - 2 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 4 T^{2} + 46 T^{3} + 6 T^{4} - 230 T^{5} + 1699 T^{6} - 230 p T^{7} + 6 p^{2} T^{8} + 46 p^{3} T^{9} - 4 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 3 T^{2} - 84 T^{3} - 15 p T^{4} + 345 T^{5} + 5006 T^{6} + 345 p T^{7} - 15 p^{3} T^{8} - 84 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T - 28 T^{2} + 22 T^{3} + 438 T^{4} - 926 T^{5} - 8297 T^{6} - 926 p T^{7} + 438 p^{2} T^{8} + 22 p^{3} T^{9} - 28 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 24 T^{2} + 29 T^{3} + 357 T^{4} - 1524 T^{5} - 8997 T^{6} - 1524 p T^{7} + 357 p^{2} T^{8} + 29 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 14 T + 74 T^{2} + 358 T^{3} + 2628 T^{4} + 11188 T^{5} + 33943 T^{6} + 11188 p T^{7} + 2628 p^{2} T^{8} + 358 p^{3} T^{9} + 74 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + T - 46 T^{2} + 149 T^{3} + 897 T^{4} - 4282 T^{5} - 13523 T^{6} - 4282 p T^{7} + 897 p^{2} T^{8} + 149 p^{3} T^{9} - 46 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 3 T + 57 T^{2} + 159 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 72 T^{2} + 155 T^{3} + 2967 T^{4} - 2244 T^{5} - 114171 T^{6} - 2244 p T^{7} + 2967 p^{2} T^{8} + 155 p^{3} T^{9} - 72 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 14820 p T^{7} - 1947 p^{2} T^{8} - 979 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 21 T + 261 T^{2} - 2181 T^{3} + 261 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 6 T - 126 T^{2} - 282 T^{3} + 13896 T^{4} + 15396 T^{5} - 801173 T^{6} + 15396 p T^{7} + 13896 p^{2} T^{8} - 282 p^{3} T^{9} - 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 31 T + 485 T^{2} - 4647 T^{3} + 485 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T - 12 T^{2} + 357 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 6 T + 186 T^{2} - 811 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 17 T + 119 T^{2} - 507 T^{3} + 119 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 186 T^{2} + 133 T^{3} + 22713 T^{4} - 582 T^{5} - 1916871 T^{6} - 582 p T^{7} + 22713 p^{2} T^{8} + 133 p^{3} T^{9} - 186 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 9 T + 195 T^{2} + 1053 T^{3} + 195 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 20 T + 38 T^{2} + 346 T^{3} + 32058 T^{4} + 183754 T^{5} - 606869 T^{6} + 183754 p T^{7} + 32058 p^{2} T^{8} + 346 p^{3} T^{9} + 38 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T - 72 T^{2} - 258 T^{3} + 10332 T^{4} - 58524 T^{5} - 1852445 T^{6} - 58524 p T^{7} + 10332 p^{2} T^{8} - 258 p^{3} T^{9} - 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 73950 p T^{7} - 7707 p^{2} T^{8} + 2023 p^{3} T^{9} - 66 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.01303528503524086140536540735, −4.50135526802948469655068614655, −4.41530156844755086825077920998, −4.23968076682299230552723010106, −4.21687959377831319876018017360, −4.02536913713606783732089852095, −3.83336612247930154915322841243, −3.69532555717274829540187857271, −3.67185647597434142689330037599, −3.52342424899645714582401170086, −3.41109568997960740313877013525, −3.02042751107858000578435070866, −2.64946879005486828028889866571, −2.64197972654556349550498302896, −2.48428843006997873869944111069, −2.38915201023572186298100851008, −2.35291259163629691458704836445, −2.32647919953457748437649639217, −2.05774360937792651331486295428, −1.63639639429373277654652417267, −1.52123844052684382105006510488, −1.06843623288592880597873195408, −0.70199318666124335369437830233, −0.57655025910774539293761534121, −0.48543406131014476757539569819, 0.48543406131014476757539569819, 0.57655025910774539293761534121, 0.70199318666124335369437830233, 1.06843623288592880597873195408, 1.52123844052684382105006510488, 1.63639639429373277654652417267, 2.05774360937792651331486295428, 2.32647919953457748437649639217, 2.35291259163629691458704836445, 2.38915201023572186298100851008, 2.48428843006997873869944111069, 2.64197972654556349550498302896, 2.64946879005486828028889866571, 3.02042751107858000578435070866, 3.41109568997960740313877013525, 3.52342424899645714582401170086, 3.67185647597434142689330037599, 3.69532555717274829540187857271, 3.83336612247930154915322841243, 4.02536913713606783732089852095, 4.21687959377831319876018017360, 4.23968076682299230552723010106, 4.41530156844755086825077920998, 4.50135526802948469655068614655, 5.01303528503524086140536540735

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

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