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Properties

Label 12-42e12-1.1-c1e6-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 4·11-s − 3·13-s − 4·15-s − 2·17-s − 3·19-s + 28·23-s − 19·25-s + 5·27-s − 29-s + 3·31-s − 8·33-s + 3·37-s + 6·39-s − 3·43-s − 21·47-s + 4·51-s − 6·53-s + 8·55-s + 6·57-s − 31·59-s − 6·61-s − 6·65-s − 6·67-s − 56·69-s + 34·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 1.20·11-s − 0.832·13-s − 1.03·15-s − 0.485·17-s − 0.688·19-s + 5.83·23-s − 3.79·25-s + 0.962·27-s − 0.185·29-s + 0.538·31-s − 1.39·33-s + 0.493·37-s + 0.960·39-s − 0.457·43-s − 3.06·47-s + 0.560·51-s − 0.824·53-s + 1.07·55-s + 0.794·57-s − 4.03·59-s − 0.768·61-s − 0.744·65-s − 0.733·67-s − 6.74·69-s + 4.03·71-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

\(L(1)\) \(\approx\) \(0.4387935466\)
\(L(\frac12)\) \(\approx\) \(0.4387935466\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + 4 T^{2} + p T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good5 \( ( 1 - T + 11 T^{2} - 9 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 2 T + 8 T^{2} + 15 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 14 T + 122 T^{2} - 675 T^{3} + 122 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 1344 p T^{7} + 1005 p^{2} T^{8} + 147 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 21 T + 180 T^{2} + 1119 T^{3} + 10053 T^{4} + 100416 T^{5} + 788551 T^{6} + 100416 p T^{7} + 10053 p^{2} T^{8} + 1119 p^{3} T^{9} + 180 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 31 T + 476 T^{2} + 5741 T^{3} + 62553 T^{4} + 587576 T^{5} + 4781851 T^{6} + 587576 p T^{7} + 62553 p^{2} T^{8} + 5741 p^{3} T^{9} + 476 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 35394 p T^{7} + 3018 p^{2} T^{8} + 642 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 28236 p T^{7} + 17268 p^{2} T^{8} - 506 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 15786 p T^{7} + 13143 p^{2} T^{8} + 351 p^{3} T^{9} - 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.86588103824702198021984441636, −4.75361244316340053019677861225, −4.66018553147435605008230952404, −4.55349993043153232654949358494, −4.39841143714459582083196683004, −4.08870045045317023389044762523, −3.91861787381910167085290305347, −3.71756837355833329631885867910, −3.63158249702633270570841066615, −3.52223094277898337708427410456, −3.19907065032058515904256941259, −2.99147052470077674715453490391, −2.83102669147968056581372248416, −2.82857764387531685927367347983, −2.74063773561527040677665358032, −2.40920223524449597790810172069, −1.99040980158979522600478008067, −1.90719618354337062399702349353, −1.83869968476673478706946277266, −1.46025163921275557338660898885, −1.39102822775476574489128543932, −1.13462845441402703929232857476, −0.929700252686879874376443066466, −0.41994846749220256501017049490, −0.11913600292712082133281908758, 0.11913600292712082133281908758, 0.41994846749220256501017049490, 0.929700252686879874376443066466, 1.13462845441402703929232857476, 1.39102822775476574489128543932, 1.46025163921275557338660898885, 1.83869968476673478706946277266, 1.90719618354337062399702349353, 1.99040980158979522600478008067, 2.40920223524449597790810172069, 2.74063773561527040677665358032, 2.82857764387531685927367347983, 2.83102669147968056581372248416, 2.99147052470077674715453490391, 3.19907065032058515904256941259, 3.52223094277898337708427410456, 3.63158249702633270570841066615, 3.71756837355833329631885867910, 3.91861787381910167085290305347, 4.08870045045317023389044762523, 4.39841143714459582083196683004, 4.55349993043153232654949358494, 4.66018553147435605008230952404, 4.75361244316340053019677861225, 4.86588103824702198021984441636

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

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