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Properties

Label 12-43e6-1.1-c1e6-0-0
Degree $12$
Conductor $6321363049$
Sign $1$
Analytic cond. $0.00163860$
Root an. cond. $0.585966$
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  − 5·2-s + 2·3-s + 16·4-s − 3·5-s − 10·6-s − 42·8-s + 3·9-s + 15·10-s − 10·11-s + 32·12-s + 5·13-s − 6·15-s + 98·16-s − 4·17-s − 15·18-s + 2·19-s − 48·20-s + 50·22-s + 23-s − 84·24-s + 5·25-s − 25·26-s + 6·29-s + 30·30-s − 6·31-s − 210·32-s − 20·33-s + ⋯
L(s)  = 1  − 3.53·2-s + 1.15·3-s + 8·4-s − 1.34·5-s − 4.08·6-s − 14.8·8-s + 9-s + 4.74·10-s − 3.01·11-s + 9.23·12-s + 1.38·13-s − 1.54·15-s + 49/2·16-s − 0.970·17-s − 3.53·18-s + 0.458·19-s − 10.7·20-s + 10.6·22-s + 0.208·23-s − 17.1·24-s + 25-s − 4.90·26-s + 1.11·29-s + 5.47·30-s − 1.07·31-s − 37.1·32-s − 3.48·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(43^{6}\)
Sign: $1$
Analytic conductor: \(0.00163860\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 43^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.07328804383\)
\(L(\frac12)\) \(\approx\) \(0.07328804383\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + 13 T + 141 T^{2} + 895 T^{3} + 141 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 + 5 T + 9 T^{2} + 7 T^{3} + 3 T^{4} + T^{5} - T^{6} + p T^{7} + 3 p^{2} T^{8} + 7 p^{3} T^{9} + 9 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 - 2 T + T^{2} + 4 T^{3} + p T^{4} - 4 T^{5} + 13 T^{6} - 4 p T^{7} + p^{3} T^{8} + 4 p^{3} T^{9} + p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 4 T^{2} + 18 T^{3} + 76 T^{4} + 29 p T^{5} + 251 T^{6} + 29 p^{2} T^{7} + 76 p^{2} T^{8} + 18 p^{3} T^{9} + 4 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} )^{2} \)
11 \( 1 + 10 T + 26 T^{2} - 81 T^{3} - 529 T^{4} - 199 T^{5} + 3535 T^{6} - 199 p T^{7} - 529 p^{2} T^{8} - 81 p^{3} T^{9} + 26 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 5 T + 40 T^{2} - 212 T^{3} + 953 T^{4} - 4319 T^{5} + 14876 T^{6} - 4319 p T^{7} + 953 p^{2} T^{8} - 212 p^{3} T^{9} + 40 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T - T^{2} - 72 T^{3} + 93 T^{4} - 42 T^{5} - 293 T^{6} - 42 p T^{7} + 93 p^{2} T^{8} - 72 p^{3} T^{9} - p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 13 T^{2} + 40 T^{3} + 23 T^{4} + 1756 T^{5} - 1723 T^{6} + 1756 p T^{7} + 23 p^{2} T^{8} + 40 p^{3} T^{9} + 13 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - T - 8 T^{2} + 255 T^{3} - 57 T^{4} - 1090 T^{5} + 35133 T^{6} - 1090 p T^{7} - 57 p^{2} T^{8} + 255 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 2142 T^{5} + 16003 T^{6} + 2142 p T^{7} - 995 p^{2} T^{8} + 132 p^{3} T^{9} + 7 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 6 T - 23 T^{2} - 478 T^{3} - 1763 T^{4} + 9154 T^{5} + 96655 T^{6} + 9154 p T^{7} - 1763 p^{2} T^{8} - 478 p^{3} T^{9} - 23 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 7 T + 97 T^{2} + 427 T^{3} + 97 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 2 T + 19 T^{2} + 30 T^{3} - 307 T^{4} - 98 p T^{5} + 1483 p T^{6} - 98 p^{2} T^{7} - 307 p^{2} T^{8} + 30 p^{3} T^{9} + 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 17 T + 158 T^{2} - 816 T^{3} + 3471 T^{4} - 309 p T^{5} + 108772 T^{6} - 309 p^{2} T^{7} + 3471 p^{2} T^{8} - 816 p^{3} T^{9} + 158 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 2 T + 56 T^{2} + 531 T^{3} + 6599 T^{4} + 27685 T^{5} + 387913 T^{6} + 27685 p T^{7} + 6599 p^{2} T^{8} + 531 p^{3} T^{9} + 56 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T - 20 T^{2} + 1053 T^{3} - 3791 T^{4} - 30145 T^{5} + 385979 T^{6} - 30145 p T^{7} - 3791 p^{2} T^{8} + 1053 p^{3} T^{9} - 20 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T + 27 T^{2} - 404 T^{3} + 3229 T^{4} + 16766 T^{5} - 226185 T^{6} + 16766 p T^{7} + 3229 p^{2} T^{8} - 404 p^{3} T^{9} + 27 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 117 T^{2} + 704 T^{3} + 12085 T^{4} + 129244 T^{5} + 998865 T^{6} + 129244 p T^{7} + 12085 p^{2} T^{8} + 704 p^{3} T^{9} + 117 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 26 T + 101 T^{2} + 3378 T^{3} - 35947 T^{4} - 149374 T^{5} + 4056325 T^{6} - 149374 p T^{7} - 35947 p^{2} T^{8} + 3378 p^{3} T^{9} + 101 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 9 T - 6 T^{2} + 430 T^{3} - 1187 T^{4} - 23719 T^{5} + 416688 T^{6} - 23719 p T^{7} - 1187 p^{2} T^{8} + 430 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 10 T + 254 T^{2} + 1581 T^{3} + 254 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 20 T + 72 T^{2} + 1865 T^{3} - 18531 T^{4} - 575 p T^{5} + 21001 p T^{6} - 575 p^{2} T^{7} - 18531 p^{2} T^{8} + 1865 p^{3} T^{9} + 72 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 11 T + 81 T^{2} + 431 T^{3} + 2694 T^{4} - 97288 T^{5} + 1979249 T^{6} - 97288 p T^{7} + 2694 p^{2} T^{8} + 431 p^{3} T^{9} + 81 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 449 T^{2} - 5950 T^{3} + 80095 T^{4} - 914662 T^{5} + 9384215 T^{6} - 914662 p T^{7} + 80095 p^{2} T^{8} - 5950 p^{3} T^{9} + 449 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−9.239100655431145950324598655377, −8.971944856428786149472239688581, −8.683375405109862792957169873101, −8.606508672414175065369680913843, −8.539075099182556760395071527711, −8.441261838996480769431665392131, −7.941318874772606490919485943225, −7.934675546452627457963937252252, −7.56226390010543175341079984875, −7.52255118476162318472367986834, −7.10419038033611714736433613675, −7.03917014221067456600053430680, −6.88255250441140821997469908598, −6.23530673000768311636743758546, −6.04124441270756831346492612690, −5.97748392060107263565624987378, −5.32565977658481425889969363294, −4.93311376787048380702474072068, −4.84318037718444795871189823885, −3.73060713755897409966563918077, −3.44152526760989267585573380952, −3.30347768089725826481993777647, −2.89561549258642634439985197996, −2.22392575225500720392722814358, −1.93492967891921699407870426510, 1.93492967891921699407870426510, 2.22392575225500720392722814358, 2.89561549258642634439985197996, 3.30347768089725826481993777647, 3.44152526760989267585573380952, 3.73060713755897409966563918077, 4.84318037718444795871189823885, 4.93311376787048380702474072068, 5.32565977658481425889969363294, 5.97748392060107263565624987378, 6.04124441270756831346492612690, 6.23530673000768311636743758546, 6.88255250441140821997469908598, 7.03917014221067456600053430680, 7.10419038033611714736433613675, 7.52255118476162318472367986834, 7.56226390010543175341079984875, 7.934675546452627457963937252252, 7.941318874772606490919485943225, 8.441261838996480769431665392131, 8.539075099182556760395071527711, 8.606508672414175065369680913843, 8.683375405109862792957169873101, 8.971944856428786149472239688581, 9.239100655431145950324598655377

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

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