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Properties

Label 12-42e12-1.1-c1e6-0-6
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  + 2·3-s + 5-s + 6·9-s − 2·11-s + 3·13-s + 2·15-s − 4·17-s − 6·19-s − 14·23-s + 11·25-s + 7·27-s − 29-s − 3·31-s − 4·33-s − 6·37-s + 6·39-s − 3·43-s + 6·45-s + 21·47-s − 8·51-s + 12·53-s − 2·55-s − 12·57-s + 31·59-s + 6·61-s + 3·65-s − 6·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 2·9-s − 0.603·11-s + 0.832·13-s + 0.516·15-s − 0.970·17-s − 1.37·19-s − 2.91·23-s + 11/5·25-s + 1.34·27-s − 0.185·29-s − 0.538·31-s − 0.696·33-s − 0.986·37-s + 0.960·39-s − 0.457·43-s + 0.894·45-s + 3.06·47-s − 1.12·51-s + 1.64·53-s − 0.269·55-s − 1.58·57-s + 4.03·59-s + 0.768·61-s + 0.372·65-s − 0.733·67-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

\(L(1)\) \(\approx\) \(16.50495745\)
\(L(\frac12)\) \(\approx\) \(16.50495745\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T - 2 T^{2} + p^{2} T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
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 \( ( 1 + 2 T + 32 T^{2} + 21 T^{3} + 32 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 33 T^{2} + 35 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 + 3 T - 48 T^{2} - 147 T^{3} + 1005 T^{4} + 1344 T^{5} - 24505 T^{6} + 1344 p T^{7} + 1005 p^{2} T^{8} - 147 p^{3} T^{9} - 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 81 T^{2} + 199 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 6 T + 162 T^{2} - 627 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 3 T + 195 T^{2} - 359 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 12 T + 216 T^{2} + 1425 T^{3} + 216 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−4.83499915051380659819745496431, −4.59111320660843624605352238204, −4.39442907023558392510257909643, −4.35751893039176934000843165490, −4.26150615003033586097701064175, −3.97419985136907493367635670964, −3.96051170032239938861343624518, −3.81511755197023790480399028953, −3.55594807015698171532085853411, −3.54131843844701919339776408986, −3.42032787559572487649908137769, −3.02147000126625590960760328083, −3.00620900138416945465463444778, −2.53347667993618828349484219316, −2.47585520508381724223350860315, −2.25506716959470395903380307495, −2.24984536710322698267771634158, −2.08125663325927905375026003887, −1.82660533984197579320146263773, −1.79663080144774905293221094850, −1.57879873901324374279941135105, −0.979330793066210489694237825233, −0.73264161206632684455143713006, −0.71687220000516924054773501633, −0.46937208633514497947741358456, 0.46937208633514497947741358456, 0.71687220000516924054773501633, 0.73264161206632684455143713006, 0.979330793066210489694237825233, 1.57879873901324374279941135105, 1.79663080144774905293221094850, 1.82660533984197579320146263773, 2.08125663325927905375026003887, 2.24984536710322698267771634158, 2.25506716959470395903380307495, 2.47585520508381724223350860315, 2.53347667993618828349484219316, 3.00620900138416945465463444778, 3.02147000126625590960760328083, 3.42032787559572487649908137769, 3.54131843844701919339776408986, 3.55594807015698171532085853411, 3.81511755197023790480399028953, 3.96051170032239938861343624518, 3.97419985136907493367635670964, 4.26150615003033586097701064175, 4.35751893039176934000843165490, 4.39442907023558392510257909643, 4.59111320660843624605352238204, 4.83499915051380659819745496431

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

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