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Properties

Label 12-7600e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.927\times 10^{23}$
Sign $1$
Analytic cond. $4.99510\times 10^{10}$
Root an. cond. $7.79014$
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 − 6·7-s − 3·9-s + 2·11-s + 14·13-s + 10·17-s − 6·19-s + 12·21-s − 2·23-s + 10·27-s − 2·29-s − 8·31-s − 4·33-s + 4·37-s − 28·39-s + 4·41-s − 4·43-s − 4·47-s − 2·49-s − 20·51-s − 14·53-s + 12·57-s + 2·59-s + 10·61-s + 18·63-s − 42·67-s + 4·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.26·7-s − 9-s + 0.603·11-s + 3.88·13-s + 2.42·17-s − 1.37·19-s + 2.61·21-s − 0.417·23-s + 1.92·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.657·37-s − 4.48·39-s + 0.624·41-s − 0.609·43-s − 0.583·47-s − 2/7·49-s − 2.80·51-s − 1.92·53-s + 1.58·57-s + 0.260·59-s + 1.28·61-s + 2.26·63-s − 5.13·67-s + 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(4.99510\times 10^{10}\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.751031266\)
\(L(\frac12)\) \(\approx\) \(2.751031266\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 25 T^{4} + 32 T^{5} + 86 T^{6} + 32 p T^{7} + 25 p^{2} T^{8} + 10 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 38 T^{2} + 148 T^{3} + 82 p T^{4} + 1674 T^{5} + 4994 T^{6} + 1674 p T^{7} + 82 p^{3} T^{8} + 148 p^{3} T^{9} + 38 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T + 13 T^{2} - 34 T^{3} + 245 T^{4} - 368 T^{5} + 2322 T^{6} - 368 p T^{7} + 245 p^{2} T^{8} - 34 p^{3} T^{9} + 13 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 14 T + 105 T^{2} - 558 T^{3} + 2669 T^{4} - 924 p T^{5} + 47674 T^{6} - 924 p^{2} T^{7} + 2669 p^{2} T^{8} - 558 p^{3} T^{9} + 105 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 10 T + 4 p T^{2} - 276 T^{3} + 1308 T^{4} - 6202 T^{5} + 30858 T^{6} - 6202 p T^{7} + 1308 p^{2} T^{8} - 276 p^{3} T^{9} + 4 p^{5} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T + 77 T^{2} + 186 T^{3} + 2931 T^{4} + 6760 T^{5} + 77966 T^{6} + 6760 p T^{7} + 2931 p^{2} T^{8} + 186 p^{3} T^{9} + 77 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 2 T + 79 T^{2} - 102 T^{3} + 3027 T^{4} - 8324 T^{5} + 101530 T^{6} - 8324 p T^{7} + 3027 p^{2} T^{8} - 102 p^{3} T^{9} + 79 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 148 T^{2} + 920 T^{3} + 10167 T^{4} + 50192 T^{5} + 403176 T^{6} + 50192 p T^{7} + 10167 p^{2} T^{8} + 920 p^{3} T^{9} + 148 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T + 100 T^{2} - 436 T^{3} + 6119 T^{4} - 22616 T^{5} + 267960 T^{6} - 22616 p T^{7} + 6119 p^{2} T^{8} - 436 p^{3} T^{9} + 100 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 188 T^{2} - 676 T^{3} + 16775 T^{4} - 50488 T^{5} + 876504 T^{6} - 50488 p T^{7} + 16775 p^{2} T^{8} - 676 p^{3} T^{9} + 188 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 4 T + 141 T^{2} + 712 T^{3} + 11185 T^{4} + 49420 T^{5} + 594874 T^{6} + 49420 p T^{7} + 11185 p^{2} T^{8} + 712 p^{3} T^{9} + 141 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 4 T + 243 T^{2} + 816 T^{3} + 26239 T^{4} + 71836 T^{5} + 1599386 T^{6} + 71836 p T^{7} + 26239 p^{2} T^{8} + 816 p^{3} T^{9} + 243 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 14 T + 249 T^{2} + 2462 T^{3} + 28173 T^{4} + 220540 T^{5} + 1887962 T^{6} + 220540 p T^{7} + 28173 p^{2} T^{8} + 2462 p^{3} T^{9} + 249 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 89 T^{2} - 114 T^{3} + 9599 T^{4} - 8120 T^{5} + 549326 T^{6} - 8120 p T^{7} + 9599 p^{2} T^{8} - 114 p^{3} T^{9} + 89 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 10 T + 233 T^{2} - 1710 T^{3} + 27173 T^{4} - 171840 T^{5} + 2075338 T^{6} - 171840 p T^{7} + 27173 p^{2} T^{8} - 1710 p^{3} T^{9} + 233 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 42 T + 1067 T^{2} + 18970 T^{3} + 262101 T^{4} + 2890456 T^{5} + 26146286 T^{6} + 2890456 p T^{7} + 262101 p^{2} T^{8} + 18970 p^{3} T^{9} + 1067 p^{4} T^{10} + 42 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 8 T + 364 T^{2} - 2736 T^{3} + 58431 T^{4} - 381256 T^{5} + 5343112 T^{6} - 381256 p T^{7} + 58431 p^{2} T^{8} - 2736 p^{3} T^{9} + 364 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 276 T^{2} + 1356 T^{3} + 33692 T^{4} + 238482 T^{5} + 2758666 T^{6} + 238482 p T^{7} + 33692 p^{2} T^{8} + 1356 p^{3} T^{9} + 276 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 20 T + 590 T^{2} - 7996 T^{3} + 129599 T^{4} - 1270184 T^{5} + 14099364 T^{6} - 1270184 p T^{7} + 129599 p^{2} T^{8} - 7996 p^{3} T^{9} + 590 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 16 T + 510 T^{2} + 6064 T^{3} + 105831 T^{4} + 961376 T^{5} + 11690948 T^{6} + 961376 p T^{7} + 105831 p^{2} T^{8} + 6064 p^{3} T^{9} + 510 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 32 T + 656 T^{2} - 9816 T^{3} + 119487 T^{4} - 1272488 T^{5} + 12355776 T^{6} - 1272488 p T^{7} + 119487 p^{2} T^{8} - 9816 p^{3} T^{9} + 656 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 40 T + 1060 T^{2} - 20000 T^{3} + 303631 T^{4} - 3806872 T^{5} + 40585688 T^{6} - 3806872 p T^{7} + 303631 p^{2} T^{8} - 20000 p^{3} T^{9} + 1060 p^{4} T^{10} - 40 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.85751300902908814077678678635, −3.84769437707809171421170542804, −3.65458329490301999150406232573, −3.59552663912504136115310666561, −3.49042613835237667598536335247, −3.47313269177716019902152465658, −3.39509064696964915838393710192, −2.90996476896736343228278892237, −2.87154093118922396942745745069, −2.78547272096025697743806000467, −2.72397375470421903291551426133, −2.71116027853976260235049529412, −2.65333080534119838244537247897, −2.02015301710794022246088007399, −1.85805713515105851009196496094, −1.64389101712918143231340359065, −1.62857762898533227919339308452, −1.51119498538421714094356967968, −1.50127090287958302429645447817, −1.27802663322557661767391640523, −0.854959175260131162313690302876, −0.70283306990231567103521116825, −0.45006729946314123829404837123, −0.40735182178599663148089247510, −0.24887439034358529836019497340, 0.24887439034358529836019497340, 0.40735182178599663148089247510, 0.45006729946314123829404837123, 0.70283306990231567103521116825, 0.854959175260131162313690302876, 1.27802663322557661767391640523, 1.50127090287958302429645447817, 1.51119498538421714094356967968, 1.62857762898533227919339308452, 1.64389101712918143231340359065, 1.85805713515105851009196496094, 2.02015301710794022246088007399, 2.65333080534119838244537247897, 2.71116027853976260235049529412, 2.72397375470421903291551426133, 2.78547272096025697743806000467, 2.87154093118922396942745745069, 2.90996476896736343228278892237, 3.39509064696964915838393710192, 3.47313269177716019902152465658, 3.49042613835237667598536335247, 3.59552663912504136115310666561, 3.65458329490301999150406232573, 3.84769437707809171421170542804, 3.85751300902908814077678678635

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

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