Properties

Label 12-912e6-1.1-c1e6-0-8
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  − 6·5-s + 9·7-s + 9·11-s − 6·13-s − 12·17-s + 18·19-s + 3·23-s + 18·25-s + 27-s + 6·31-s − 54·35-s − 12·41-s + 39·47-s + 27·49-s − 12·53-s − 54·55-s + 12·59-s + 27·61-s + 36·65-s + 36·67-s + 18·71-s − 9·73-s + 81·77-s − 18·79-s − 9·83-s + 72·85-s + 3·89-s + ⋯
L(s)  = 1  − 2.68·5-s + 3.40·7-s + 2.71·11-s − 1.66·13-s − 2.91·17-s + 4.12·19-s + 0.625·23-s + 18/5·25-s + 0.192·27-s + 1.07·31-s − 9.12·35-s − 1.87·41-s + 5.68·47-s + 27/7·49-s − 1.64·53-s − 7.28·55-s + 1.56·59-s + 3.45·61-s + 4.46·65-s + 4.39·67-s + 2.13·71-s − 1.05·73-s + 9.23·77-s − 2.02·79-s − 0.987·83-s + 7.80·85-s + 0.317·89-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

\(L(1)\) \(\approx\) \(3.867231517\)
\(L(\frac12)\) \(\approx\) \(3.867231517\)
\(L(\frac{3}{2})\) 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 \( 1 - T^{3} + T^{6} \)
19 \( 1 - 18 T + 144 T^{2} - 737 T^{3} + 144 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 9 T + 54 T^{2} - 243 T^{3} + 927 T^{4} - 3042 T^{5} + 8641 T^{6} - 3042 p T^{7} + 927 p^{2} T^{8} - 243 p^{3} T^{9} + 54 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 9 T + 60 T^{2} - 27 p T^{3} + 1239 T^{4} - 414 p T^{5} + 15797 T^{6} - 414 p^{2} T^{7} + 1239 p^{2} T^{8} - 27 p^{4} T^{9} + 60 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 24 T^{2} + 84 T^{3} - 42 T^{4} - 1266 T^{5} - 4703 T^{6} - 1266 p T^{7} - 42 p^{2} T^{8} + 84 p^{3} T^{9} + 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 12 T + 54 T^{2} + 81 T^{3} + 9 p T^{4} + 3999 T^{5} + 27073 T^{6} + 3999 p T^{7} + 9 p^{3} T^{8} + 81 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T + 24 T^{2} - 6 T^{3} - 39 T^{4} - 1131 T^{5} - 1033 T^{6} - 1131 p T^{7} - 39 p^{2} T^{8} - 6 p^{3} T^{9} + 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 18 T^{2} - 36 T^{3} + 1062 T^{4} - 1854 T^{5} - 18703 T^{6} - 1854 p T^{7} + 1062 p^{2} T^{8} - 36 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 - 6 T - 12 T^{2} - 82 T^{3} + 198 T^{4} + 7146 T^{5} - 35829 T^{6} + 7146 p T^{7} + 198 p^{2} T^{8} - 82 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 60 T^{2} + 3768 T^{4} - 145631 T^{6} + 3768 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 12 T - 39 T^{2} - 1443 T^{3} - 6699 T^{4} + 38607 T^{5} + 551234 T^{6} + 38607 p T^{7} - 6699 p^{2} T^{8} - 1443 p^{3} T^{9} - 39 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T^{2} - 234 T^{3} - 1854 T^{4} + 11574 T^{5} - 44261 T^{6} + 11574 p T^{7} - 1854 p^{2} T^{8} - 234 p^{3} T^{9} + 18 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 39 T + 699 T^{2} - 8097 T^{3} + 74832 T^{4} - 621420 T^{5} + 4589651 T^{6} - 621420 p T^{7} + 74832 p^{2} T^{8} - 8097 p^{3} T^{9} + 699 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 51 T^{2} - 651 T^{3} - 5583 T^{4} - 10965 T^{5} + 186374 T^{6} - 10965 p T^{7} - 5583 p^{2} T^{8} - 651 p^{3} T^{9} + 51 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 234 T^{2} - 2403 T^{3} + 22707 T^{4} - 218721 T^{5} + 1460809 T^{6} - 218721 p T^{7} + 22707 p^{2} T^{8} - 2403 p^{3} T^{9} + 234 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 27 T + 324 T^{2} - 2414 T^{3} + 12636 T^{4} - 35397 T^{5} + 18543 T^{6} - 35397 p T^{7} + 12636 p^{2} T^{8} - 2414 p^{3} T^{9} + 324 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 36 T + 576 T^{2} - 5362 T^{3} + 26892 T^{4} + 18954 T^{5} - 1159755 T^{6} + 18954 p T^{7} + 26892 p^{2} T^{8} - 5362 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 198 T^{2} - 2088 T^{3} + 21024 T^{4} - 168804 T^{5} + 1342873 T^{6} - 168804 p T^{7} + 21024 p^{2} T^{8} - 2088 p^{3} T^{9} + 198 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 9 T - 18 T^{2} + 376 T^{3} - 351 T^{4} - 5157 T^{5} + 665553 T^{6} - 5157 p T^{7} - 351 p^{2} T^{8} + 376 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 18 T + 18 T^{2} - 2360 T^{3} - 22032 T^{4} + 72252 T^{5} + 2325153 T^{6} + 72252 p T^{7} - 22032 p^{2} T^{8} - 2360 p^{3} T^{9} + 18 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 9 T + 264 T^{2} + 2133 T^{3} + 40989 T^{4} + 292788 T^{5} + 4079453 T^{6} + 292788 p T^{7} + 40989 p^{2} T^{8} + 2133 p^{3} T^{9} + 264 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T + 6 T^{2} - 2562 T^{3} + 3003 T^{4} - 8997 T^{5} + 2885051 T^{6} - 8997 p T^{7} + 3003 p^{2} T^{8} - 2562 p^{3} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T + 6 T^{2} + 1428 T^{3} - 3957 T^{4} - 93747 T^{5} + 930127 T^{6} - 93747 p T^{7} - 3957 p^{2} T^{8} + 1428 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.22828128189217263579376490752, −5.09390814196005176002295391599, −5.04083877349462980400541246632, −4.75322542716792886262385876807, −4.73709212038913243709730937534, −4.65059682982139770125851951021, −4.31014945893034408023456384499, −4.10970192531577293839517950489, −3.89783418626098069021472324981, −3.85488280150738129907943863707, −3.84827666655221956893589492222, −3.70572400550626737558793013265, −3.59609981640647122666358399276, −3.02440138084909670579583490088, −2.81243922636853451985235992483, −2.55877759232529329467915660013, −2.55519829918482498309272213770, −2.32748606940211903864412455800, −2.18245239612216882474597645900, −1.58568629662808568708608449816, −1.42452957051676808160622173652, −1.18665425438903847191621187816, −1.14357783312529414856199612189, −0.847222050758597136263426211324, −0.33195007223016863703055867194, 0.33195007223016863703055867194, 0.847222050758597136263426211324, 1.14357783312529414856199612189, 1.18665425438903847191621187816, 1.42452957051676808160622173652, 1.58568629662808568708608449816, 2.18245239612216882474597645900, 2.32748606940211903864412455800, 2.55519829918482498309272213770, 2.55877759232529329467915660013, 2.81243922636853451985235992483, 3.02440138084909670579583490088, 3.59609981640647122666358399276, 3.70572400550626737558793013265, 3.84827666655221956893589492222, 3.85488280150738129907943863707, 3.89783418626098069021472324981, 4.10970192531577293839517950489, 4.31014945893034408023456384499, 4.65059682982139770125851951021, 4.73709212038913243709730937534, 4.75322542716792886262385876807, 5.04083877349462980400541246632, 5.09390814196005176002295391599, 5.22828128189217263579376490752

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

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