Properties

Label 12-570e6-1.1-c1e6-0-1
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  + 3·7-s − 8-s + 3·11-s + 9·13-s − 9·17-s − 9·19-s − 21·23-s + 27-s − 15·29-s + 6·31-s + 6·37-s + 18·41-s − 15·43-s − 6·47-s + 21·49-s − 12·53-s − 3·56-s − 3·59-s + 9·61-s − 24·67-s − 15·73-s + 9·77-s + 27·79-s − 9·83-s − 3·88-s − 33·89-s + 27·91-s + ⋯
L(s)  = 1  + 1.13·7-s − 0.353·8-s + 0.904·11-s + 2.49·13-s − 2.18·17-s − 2.06·19-s − 4.37·23-s + 0.192·27-s − 2.78·29-s + 1.07·31-s + 0.986·37-s + 2.81·41-s − 2.28·43-s − 0.875·47-s + 3·49-s − 1.64·53-s − 0.400·56-s − 0.390·59-s + 1.15·61-s − 2.93·67-s − 1.75·73-s + 1.02·77-s + 3.03·79-s − 0.987·83-s − 0.319·88-s − 3.49·89-s + 2.83·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.003685745050\)
\(L(\frac12)\) \(\approx\) \(0.003685745050\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
3 \( 1 - T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( 1 + 9 T + 72 T^{2} + 341 T^{3} + 72 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 - 3 T - 12 T^{2} + 15 T^{3} + 177 T^{4} - 12 p T^{5} - 1321 T^{6} - 12 p^{2} T^{7} + 177 p^{2} T^{8} + 15 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T - 24 T^{2} + 31 T^{3} + 531 T^{4} - 30 p T^{5} - 6181 T^{6} - 30 p^{2} T^{7} + 531 p^{2} T^{8} + 31 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 106 T^{3} + 27 T^{4} + 1161 T^{5} - 4731 T^{6} + 1161 p T^{7} + 27 p^{2} T^{8} - 106 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 18 T^{2} - 130 T^{3} - 747 T^{4} + 27 p T^{5} + 12581 T^{6} + 27 p^{2} T^{7} - 747 p^{2} T^{8} - 130 p^{3} T^{9} + 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 21 T + 201 T^{2} + 1213 T^{3} + 5448 T^{4} + 20502 T^{5} + 83381 T^{6} + 20502 p T^{7} + 5448 p^{2} T^{8} + 1213 p^{3} T^{9} + 201 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 135 T^{2} + 927 T^{3} + 6660 T^{4} + 43332 T^{5} + 255781 T^{6} + 43332 p T^{7} + 6660 p^{2} T^{8} + 927 p^{3} T^{9} + 135 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 6 T - 42 T^{2} + 238 T^{3} + 1548 T^{4} - 4284 T^{5} - 39009 T^{6} - 4284 p T^{7} + 1548 p^{2} T^{8} + 238 p^{3} T^{9} - 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 3 T + 3 p T^{2} - 219 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 18 T + 144 T^{2} - 743 T^{3} + 1899 T^{4} + 13851 T^{5} - 177427 T^{6} + 13851 p T^{7} + 1899 p^{2} T^{8} - 743 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 84 T^{2} + 2 p T^{3} - 1044 T^{4} - 4365 T^{5} - 23619 T^{6} - 4365 p T^{7} - 1044 p^{2} T^{8} + 2 p^{4} T^{9} + 84 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T - 54 T^{2} - 675 T^{3} - 2925 T^{4} + 20931 T^{5} + 339805 T^{6} + 20931 p T^{7} - 2925 p^{2} T^{8} - 675 p^{3} T^{9} - 54 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 99 T^{2} + 855 T^{3} + 7965 T^{4} + 56001 T^{5} + 369838 T^{6} + 56001 p T^{7} + 7965 p^{2} T^{8} + 855 p^{3} T^{9} + 99 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 3 T + 36 T^{2} - 18 T^{3} + 1980 T^{4} - 29805 T^{5} - 75275 T^{6} - 29805 p T^{7} + 1980 p^{2} T^{8} - 18 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T + 36 T^{2} + 250 T^{3} - 1845 T^{4} + 31329 T^{5} - 278043 T^{6} + 31329 p T^{7} - 1845 p^{2} T^{8} + 250 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 24 T + 186 T^{2} - 215 T^{3} - 10269 T^{4} - 6093 T^{5} + 534753 T^{6} - 6093 p T^{7} - 10269 p^{2} T^{8} - 215 p^{3} T^{9} + 186 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 54 T^{2} + 293 T^{3} + 2187 T^{4} - 6993 T^{5} + 343457 T^{6} - 6993 p T^{7} + 2187 p^{2} T^{8} + 293 p^{3} T^{9} + 54 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 15 T + 126 T^{2} + 402 T^{3} + 3609 T^{4} + 77109 T^{5} + 1143557 T^{6} + 77109 p T^{7} + 3609 p^{2} T^{8} + 402 p^{3} T^{9} + 126 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 27 T + 351 T^{2} - 3337 T^{3} + 19872 T^{4} - 49734 T^{5} + 58989 T^{6} - 49734 p T^{7} + 19872 p^{2} T^{8} - 3337 p^{3} T^{9} + 351 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 9 T + 30 T^{2} + 1017 T^{3} + 879 T^{4} - 28422 T^{5} + 447523 T^{6} - 28422 p T^{7} + 879 p^{2} T^{8} + 1017 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 33 T + 384 T^{2} + 1018 T^{3} - 10452 T^{4} + 65169 T^{5} + 2238815 T^{6} + 65169 p T^{7} - 10452 p^{2} T^{8} + 1018 p^{3} T^{9} + 384 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 6 T + 312 T^{2} - 1558 T^{3} + 52830 T^{4} - 223002 T^{5} + 6228999 T^{6} - 223002 p T^{7} + 52830 p^{2} T^{8} - 1558 p^{3} T^{9} + 312 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.86100626602570387874992370843, −5.72030487054198995948211613008, −5.69053152273040528359961803401, −5.49851208904861705538393696333, −4.87299156287759631376312015327, −4.67150883640759677114843653405, −4.61785448965249594221974898318, −4.42839003622035276508453495277, −4.39885244447945918953071918774, −4.25601874714169687916954228697, −3.97633248416355936004184759879, −3.76892810862553527844065442750, −3.75520005866464573697450137507, −3.56532544479615363107404416225, −3.16621267200114334095285781343, −3.03233851141583655801700967586, −2.69378939598072108606621133740, −2.27455410112133623969844119438, −2.07660420381021449072933724231, −2.02371678050525127963166875003, −1.90158095164971617548346184282, −1.65783100539560056347281473378, −1.23317445215609782005106968270, −0.881823233879708215248869544484, −0.01228722338216915245269650829, 0.01228722338216915245269650829, 0.881823233879708215248869544484, 1.23317445215609782005106968270, 1.65783100539560056347281473378, 1.90158095164971617548346184282, 2.02371678050525127963166875003, 2.07660420381021449072933724231, 2.27455410112133623969844119438, 2.69378939598072108606621133740, 3.03233851141583655801700967586, 3.16621267200114334095285781343, 3.56532544479615363107404416225, 3.75520005866464573697450137507, 3.76892810862553527844065442750, 3.97633248416355936004184759879, 4.25601874714169687916954228697, 4.39885244447945918953071918774, 4.42839003622035276508453495277, 4.61785448965249594221974898318, 4.67150883640759677114843653405, 4.87299156287759631376312015327, 5.49851208904861705538393696333, 5.69053152273040528359961803401, 5.72030487054198995948211613008, 5.86100626602570387874992370843

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

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