Properties

Label 12-912e6-1.1-c1e6-0-15
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  + 3·3-s − 2·5-s + 3·9-s − 3·13-s − 6·15-s − 2·17-s + 4·19-s − 12·23-s + 7·25-s − 2·27-s + 6·29-s − 2·31-s − 9·39-s − 12·41-s + 33·43-s − 6·45-s + 18·47-s + 17·49-s − 6·51-s + 36·53-s + 12·57-s − 2·59-s + 3·61-s + 6·65-s − 19·67-s − 36·69-s + 16·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 9-s − 0.832·13-s − 1.54·15-s − 0.485·17-s + 0.917·19-s − 2.50·23-s + 7/5·25-s − 0.384·27-s + 1.11·29-s − 0.359·31-s − 1.44·39-s − 1.87·41-s + 5.03·43-s − 0.894·45-s + 2.62·47-s + 17/7·49-s − 0.840·51-s + 4.94·53-s + 1.58·57-s − 0.260·59-s + 0.384·61-s + 0.744·65-s − 2.32·67-s − 4.33·69-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.224926722\)
\(L(\frac12)\) \(\approx\) \(8.224926722\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T + T^{2} )^{3} \)
19 \( 1 - 4 T + 17 T^{2} - 136 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 34 p T^{7} - 2 p^{2} T^{8} - 2 p^{3} T^{9} - 3 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 17 T^{2} + 198 T^{4} - 1549 T^{6} + 198 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 50 T^{2} + 1175 T^{4} - 16364 T^{6} + 1175 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 3 T + 19 T^{2} + 48 T^{3} + 33 T^{4} - 747 T^{5} - 1498 T^{6} - 747 p T^{7} + 33 p^{2} T^{8} + 48 p^{3} T^{9} + 19 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T - 27 T^{2} - 26 T^{3} + 346 T^{4} - 166 T^{5} - 5731 T^{6} - 166 p T^{7} + 346 p^{2} T^{8} - 26 p^{3} T^{9} - 27 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2666 T^{4} + 10704 T^{5} + 45865 T^{6} + 10704 p T^{7} + 2666 p^{2} T^{8} + 588 p^{3} T^{9} + 97 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 79 T^{2} - 402 T^{3} + 3290 T^{4} - 19878 T^{5} + 117439 T^{6} - 19878 p T^{7} + 3290 p^{2} T^{8} - 402 p^{3} T^{9} + 79 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + T + 84 T^{2} + 65 T^{3} + 84 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 5 T^{2} + 3390 T^{4} - 9793 T^{6} + 3390 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 12 T + 75 T^{2} + 324 T^{3} + 102 T^{4} - 16932 T^{5} - 183841 T^{6} - 16932 p T^{7} + 102 p^{2} T^{8} + 324 p^{3} T^{9} + 75 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 33 T + 585 T^{2} - 7326 T^{3} + 71847 T^{4} - 587001 T^{5} + 4129198 T^{6} - 587001 p T^{7} + 71847 p^{2} T^{8} - 7326 p^{3} T^{9} + 585 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 157 T^{2} - 882 T^{3} + 1466 T^{4} + 28590 T^{5} - 316043 T^{6} + 28590 p T^{7} + 1466 p^{2} T^{8} - 882 p^{3} T^{9} + 157 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 36 T + 727 T^{2} - 10620 T^{3} + 123086 T^{4} - 1169736 T^{5} + 9287371 T^{6} - 1169736 p T^{7} + 123086 p^{2} T^{8} - 10620 p^{3} T^{9} + 727 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T - 165 T^{2} - 110 T^{3} + 18142 T^{4} + 4934 T^{5} - 1238089 T^{6} + 4934 p T^{7} + 18142 p^{2} T^{8} - 110 p^{3} T^{9} - 165 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 33 T^{2} - 992 T^{3} + 501 T^{4} + 22395 T^{5} + 500358 T^{6} + 22395 p T^{7} + 501 p^{2} T^{8} - 992 p^{3} T^{9} - 33 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 19 T + 77 T^{2} + 102 T^{3} + 11335 T^{4} + 113975 T^{5} + 583654 T^{6} + 113975 p T^{7} + 11335 p^{2} T^{8} + 102 p^{3} T^{9} + 77 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 16 T + 3 T^{2} + 400 T^{3} + 10462 T^{4} - 68512 T^{5} - 173137 T^{6} - 68512 p T^{7} + 10462 p^{2} T^{8} + 400 p^{3} T^{9} + 3 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 11 T - 53 T^{2} + 932 T^{3} + 1921 T^{4} - 26777 T^{5} - 51994 T^{6} - 26777 p T^{7} + 1921 p^{2} T^{8} + 932 p^{3} T^{9} - 53 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 9 T - 99 T^{2} - 446 T^{3} + 8883 T^{4} - 13851 T^{5} - 1011954 T^{6} - 13851 p T^{7} + 8883 p^{2} T^{8} - 446 p^{3} T^{9} - 99 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 366 T^{2} + 63975 T^{4} - 6705124 T^{6} + 63975 p^{2} T^{8} - 366 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 6 T + 139 T^{2} - 762 T^{3} + 8570 T^{4} - 190218 T^{5} + 1098619 T^{6} - 190218 p T^{7} + 8570 p^{2} T^{8} - 762 p^{3} T^{9} + 139 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 24 T + 499 T^{2} + 7368 T^{3} + 98262 T^{4} + 1070952 T^{5} + 11457671 T^{6} + 1070952 p T^{7} + 98262 p^{2} T^{8} + 7368 p^{3} T^{9} + 499 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.40414628571157198596023438229, −5.34603602930912444559983602397, −5.07365454419974614796654135636, −4.81450083628724642924597460536, −4.45619374130239820141823887542, −4.36397436719578331695102106859, −4.25077285515649905344936071940, −4.09318778451537032527630377959, −4.05835543853070795246952998353, −3.89650819724733903516832325188, −3.66831444565740647259567135577, −3.52634388155989796220498972618, −3.37403017923393060480696825084, −2.89657813202755012730545361237, −2.75808479764963795136379112770, −2.60473091341693368255464569556, −2.57931515889451875789821064976, −2.38796607972640679783144273192, −2.36938100713938550604267800246, −1.83710953731770869445486813751, −1.76357020232966952865487311383, −1.34863762532301722798065003016, −0.77343962202013966290894783796, −0.75830768734567576895811671950, −0.51927197063057572371630986262, 0.51927197063057572371630986262, 0.75830768734567576895811671950, 0.77343962202013966290894783796, 1.34863762532301722798065003016, 1.76357020232966952865487311383, 1.83710953731770869445486813751, 2.36938100713938550604267800246, 2.38796607972640679783144273192, 2.57931515889451875789821064976, 2.60473091341693368255464569556, 2.75808479764963795136379112770, 2.89657813202755012730545361237, 3.37403017923393060480696825084, 3.52634388155989796220498972618, 3.66831444565740647259567135577, 3.89650819724733903516832325188, 4.05835543853070795246952998353, 4.09318778451537032527630377959, 4.25077285515649905344936071940, 4.36397436719578331695102106859, 4.45619374130239820141823887542, 4.81450083628724642924597460536, 5.07365454419974614796654135636, 5.34603602930912444559983602397, 5.40414628571157198596023438229

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

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