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Properties

Label 12-1134e6-1.1-c1e6-0-3
Degree $12$
Conductor $2.127\times 10^{18}$
Sign $1$
Analytic cond. $551240.$
Root an. cond. $3.00915$
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  − 3·2-s + 3·4-s − 5·5-s − 2·7-s + 2·8-s + 15·10-s − 11-s + 4·13-s + 6·14-s − 9·16-s − 4·17-s − 3·19-s − 15·20-s + 3·22-s − 7·23-s + 19·25-s − 12·26-s − 6·28-s + 10·29-s − 14·31-s + 9·32-s + 12·34-s + 10·35-s − 9·37-s + 9·38-s − 10·40-s + 24·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 2.23·5-s − 0.755·7-s + 0.707·8-s + 4.74·10-s − 0.301·11-s + 1.10·13-s + 1.60·14-s − 9/4·16-s − 0.970·17-s − 0.688·19-s − 3.35·20-s + 0.639·22-s − 1.45·23-s + 19/5·25-s − 2.35·26-s − 1.13·28-s + 1.85·29-s − 2.51·31-s + 1.59·32-s + 2.05·34-s + 1.69·35-s − 1.47·37-s + 1.45·38-s − 1.58·40-s + 3.74·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{24} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(551240.\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2891763130\)
\(L(\frac12)\) \(\approx\) \(0.2891763130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
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 \( 1 + p T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 68 p T^{7} + 31 p^{2} T^{8} + p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 32 p T^{5} + 457 p T^{6} + 32 p^{2} T^{7} - 83 p^{2} T^{8} - 103 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 2 T + 36 T^{2} - 49 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 3484 p T^{7} + 1417 p^{2} T^{8} - 127 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 5 T + 55 T^{2} - 323 T^{3} + 55 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 14 T + 58 T^{2} + 250 T^{3} + 2992 T^{4} + 9728 T^{5} - 11857 T^{6} + 9728 p T^{7} + 2992 p^{2} T^{8} + 250 p^{3} T^{9} + 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 12 T + 162 T^{2} - 1011 T^{3} + 162 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 18 T + 210 T^{2} + 1549 T^{3} + 210 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 9786 p T^{7} + 7263 p^{2} T^{8} + 267 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 26334 p T^{7} - 1179 p^{2} T^{8} + 873 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 464 p T^{5} + 7381 p T^{6} - 464 p^{2} T^{7} - 1304 p^{2} T^{8} + 994 p^{3} T^{9} - 60 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 19532 p T^{7} + 292 p^{2} T^{8} - 650 p^{3} T^{9} - 32 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 8 T + 244 T^{2} + 1235 T^{3} + 244 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 28 T + 527 T^{2} - 5968 T^{3} + 527 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.19519306690671494788902375234, −4.96299467513367235472222762136, −4.79829197923494158258656653615, −4.66488255065249405452164573364, −4.58955964254241696566197830586, −4.19448576483144773271991219079, −4.01085434130402708205454344513, −4.00584817743247724348132592096, −3.96751819356929634599813853955, −3.87694579987052817482251669890, −3.43342364269552035684143509852, −3.37364665683261707268066279128, −3.08866653232063457091120657261, −3.07625436060763517994838722589, −2.88533625539755533063736273890, −2.35686388516774258797555001776, −2.27646545032197456526097968464, −2.15340400706614926263789008071, −1.95596056654158531996416925831, −1.39629836555097936674248722684, −1.33172345444547014894403688882, −1.25383323382634493220320263644, −0.49050841873831015182161904990, −0.41508531589434567343333966809, −0.39272297397854271238514644703, 0.39272297397854271238514644703, 0.41508531589434567343333966809, 0.49050841873831015182161904990, 1.25383323382634493220320263644, 1.33172345444547014894403688882, 1.39629836555097936674248722684, 1.95596056654158531996416925831, 2.15340400706614926263789008071, 2.27646545032197456526097968464, 2.35686388516774258797555001776, 2.88533625539755533063736273890, 3.07625436060763517994838722589, 3.08866653232063457091120657261, 3.37364665683261707268066279128, 3.43342364269552035684143509852, 3.87694579987052817482251669890, 3.96751819356929634599813853955, 4.00584817743247724348132592096, 4.01085434130402708205454344513, 4.19448576483144773271991219079, 4.58955964254241696566197830586, 4.66488255065249405452164573364, 4.79829197923494158258656653615, 4.96299467513367235472222762136, 5.19519306690671494788902375234

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

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