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Properties

Label 12-378e6-1.1-c1e6-0-4
Degree $12$
Conductor $2.917\times 10^{15}$
Sign $1$
Analytic cond. $756.159$
Root an. cond. $1.73733$
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  + 6·2-s + 21·4-s − 5-s + 2·7-s + 56·8-s − 6·10-s + 11-s + 8·13-s + 12·14-s + 126·16-s + 4·17-s − 3·19-s − 21·20-s + 6·22-s + 7·23-s + 9·25-s + 48·26-s + 42·28-s + 5·29-s − 40·31-s + 252·32-s + 24·34-s − 2·35-s + 3·37-s − 18·38-s − 56·40-s − 6·43-s + ⋯
L(s)  = 1  + 4.24·2-s + 21/2·4-s − 0.447·5-s + 0.755·7-s + 19.7·8-s − 1.89·10-s + 0.301·11-s + 2.21·13-s + 3.20·14-s + 63/2·16-s + 0.970·17-s − 0.688·19-s − 4.69·20-s + 1.27·22-s + 1.45·23-s + 9/5·25-s + 9.41·26-s + 7.93·28-s + 0.928·29-s − 7.18·31-s + 44.5·32-s + 4.11·34-s − 0.338·35-s + 0.493·37-s − 2.91·38-s − 8.85·40-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(756.159\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(66.07648851\)
\(L(\frac12)\) \(\approx\) \(66.07648851\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
3 \( 1 \)
7 \( 1 - 2 T - 4 T^{2} + 31 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 52 p T^{7} + 23 p^{2} T^{8} - 17 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T - 26 T^{2} + 23 T^{3} + 37 p T^{4} - 202 T^{5} - 4853 T^{6} - 202 p T^{7} + 37 p^{3} T^{8} + 23 p^{3} T^{9} - 26 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 23 T^{2} + 4 p T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 220 p T^{7} + 410 p^{2} T^{8} + 4 p^{4} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 3946 p T^{7} + 2423 p^{2} T^{8} + 83 p^{3} T^{9} - 32 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 214 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
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 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( ( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 60360 p T^{7} + 13635 p^{2} T^{8} + 33 p^{3} T^{9} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 216 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + T + 89 T^{2} - 77 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 5 T + 163 T^{2} + 469 T^{3} + 163 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 13564 p T^{7} + 18788 p^{2} T^{8} + 2 p^{3} T^{9} - 182 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 73314 p T^{7} + 16101 p^{2} T^{8} + 1197 p^{3} T^{9} - 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.07878265916212801829735997984, −6.02501738194667865983584268696, −5.66501125091915023492950419213, −5.32562995097282969507034760154, −5.32152919242022192799525238896, −5.30677659557406930733845350369, −5.27025091890013245020229281948, −4.73029169813143067734238008664, −4.66724574913227276869726483909, −4.47046179352367386985837698871, −4.33193110203303504909978094090, −4.26923249519894237669821071818, −3.94746548860829451773895114774, −3.55586632460255318295741375386, −3.34584628962931518742568875030, −3.29951961768226093161864040708, −3.20316648994883505590536208853, −3.16534886381249362879113209807, −3.10888205063618535068314362421, −2.36429474107570020497919387375, −1.92114892921075885986819548267, −1.75608942409201619663298673545, −1.73428687845461665640461730786, −1.57367847758576328833735746604, −0.933281394689002347831166355662, 0.933281394689002347831166355662, 1.57367847758576328833735746604, 1.73428687845461665640461730786, 1.75608942409201619663298673545, 1.92114892921075885986819548267, 2.36429474107570020497919387375, 3.10888205063618535068314362421, 3.16534886381249362879113209807, 3.20316648994883505590536208853, 3.29951961768226093161864040708, 3.34584628962931518742568875030, 3.55586632460255318295741375386, 3.94746548860829451773895114774, 4.26923249519894237669821071818, 4.33193110203303504909978094090, 4.47046179352367386985837698871, 4.66724574913227276869726483909, 4.73029169813143067734238008664, 5.27025091890013245020229281948, 5.30677659557406930733845350369, 5.32152919242022192799525238896, 5.32562995097282969507034760154, 5.66501125091915023492950419213, 6.02501738194667865983584268696, 6.07878265916212801829735997984

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

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