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Properties

Label 12-1232e6-1.1-c1e6-0-6
Degree $12$
Conductor $3.497\times 10^{18}$
Sign $1$
Analytic cond. $906414.$
Root an. cond. $3.13649$
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-s + 2·5-s − 4·7-s + 3·9-s − 3·11-s + 6·13-s + 2·15-s + 5·17-s + 11·19-s − 4·21-s − 4·23-s + 4·25-s − 10·27-s − 2·29-s + 19·31-s − 3·33-s − 8·35-s + 8·37-s + 6·39-s + 34·41-s − 20·43-s + 6·45-s + 13·47-s + 2·49-s + 5·51-s − 9·53-s − 6·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 1.51·7-s + 9-s − 0.904·11-s + 1.66·13-s + 0.516·15-s + 1.21·17-s + 2.52·19-s − 0.872·21-s − 0.834·23-s + 4/5·25-s − 1.92·27-s − 0.371·29-s + 3.41·31-s − 0.522·33-s − 1.35·35-s + 1.31·37-s + 0.960·39-s + 5.30·41-s − 3.04·43-s + 0.894·45-s + 1.89·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(906414.\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.87138803\)
\(L(\frac12)\) \(\approx\) \(14.87138803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11 \( ( 1 + T + T^{2} )^{3} \)
good3 \( 1 - T - 2 T^{2} + 5 p T^{3} - p^{2} T^{4} - 2 p^{2} T^{5} + 11 p^{2} T^{6} - 2 p^{3} T^{7} - p^{4} T^{8} + 5 p^{4} T^{9} - 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T - 2 p T^{3} - 2 T^{4} + 34 T^{5} + 71 T^{6} + 34 p T^{7} - 2 p^{2} T^{8} - 2 p^{4} T^{9} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 3 T + 15 T^{2} + T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 5 T - 30 T^{2} + 59 T^{3} + 1255 T^{4} - 1280 T^{5} - 20935 T^{6} - 1280 p T^{7} + 1255 p^{2} T^{8} + 59 p^{3} T^{9} - 30 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 28 T^{2} - 121 T^{3} + 1921 T^{4} - 7040 T^{5} + 9107 T^{6} - 7040 p T^{7} + 1921 p^{2} T^{8} - 121 p^{3} T^{9} + 28 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T - 54 T^{2} - 82 T^{3} + 2566 T^{4} + 1906 T^{5} - 64069 T^{6} + 1906 p T^{7} + 2566 p^{2} T^{8} - 82 p^{3} T^{9} - 54 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + T + 37 T^{2} - 71 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 19 T + 188 T^{2} - 1125 T^{3} + 4027 T^{4} - 1478 T^{5} - 46577 T^{6} - 1478 p T^{7} + 4027 p^{2} T^{8} - 1125 p^{3} T^{9} + 188 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T - 19 T^{2} + 504 T^{3} - 854 T^{4} - 10288 T^{5} + 99853 T^{6} - 10288 p T^{7} - 854 p^{2} T^{8} + 504 p^{3} T^{9} - 19 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 17 T + 199 T^{2} - 1427 T^{3} + 199 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 10 T + 98 T^{2} + 787 T^{3} + 98 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 13 T - 24 T^{2} + 61 T^{3} + 11473 T^{4} - 40876 T^{5} - 265153 T^{6} - 40876 p T^{7} + 11473 p^{2} T^{8} + 61 p^{3} T^{9} - 24 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 6 T - 96 T^{2} + 138 T^{3} + 6846 T^{4} + 11190 T^{5} - 526349 T^{6} + 11190 p T^{7} + 6846 p^{2} T^{8} + 138 p^{3} T^{9} - 96 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 67 T^{2} + 300 T^{3} + 1762 T^{4} - 24292 T^{5} - 25547 T^{6} - 24292 p T^{7} + 1762 p^{2} T^{8} + 300 p^{3} T^{9} - 67 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 8 T - 77 T^{2} + 1160 T^{3} + 1282 T^{4} - 46064 T^{5} + 290939 T^{6} - 46064 p T^{7} + 1282 p^{2} T^{8} + 1160 p^{3} T^{9} - 77 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 13 T + 265 T^{2} - 1909 T^{3} + 265 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 22 T + 110 T^{2} + 1098 T^{3} + 34144 T^{4} + 249260 T^{5} + 697075 T^{6} + 249260 p T^{7} + 34144 p^{2} T^{8} + 1098 p^{3} T^{9} + 110 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 5 T - 80 T^{2} + 1661 T^{3} - 2735 T^{4} - 63020 T^{5} + 1153727 T^{6} - 63020 p T^{7} - 2735 p^{2} T^{8} + 1661 p^{3} T^{9} - 80 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 3 T + 153 T^{2} + 21 T^{3} + 153 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 3 T - 150 T^{2} + 1077 T^{3} + 8799 T^{4} - 62400 T^{5} - 307631 T^{6} - 62400 p T^{7} + 8799 p^{2} T^{8} + 1077 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 25 T + 347 T^{2} - 3679 T^{3} + 347 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.16158272560448824218734901506, −5.02670764644944520419100357850, −4.96458761714490718307784340808, −4.69977337626448632543905973978, −4.41286158516751917019911100796, −4.13554697444229815571300062099, −3.93609528969249748849854885758, −3.92874851580968128457985267202, −3.88210641351482324329811659325, −3.81054099366394431487385784116, −3.45353559249725831980869933792, −3.21356984783010834134914356705, −3.02182067640612861461793053916, −2.86189997625260014693917631706, −2.70598951697083816369289414866, −2.68020414429308571150446293530, −2.60690691233471318663579397865, −2.15076658063958127531879384002, −1.77674510500064850294721690132, −1.65792698040462913562445642436, −1.64688908564828840030864015019, −1.15076679071735357286544678873, −0.75113211551770793046289809912, −0.74215947158386420879381515431, −0.64550114170880522297047788687, 0.64550114170880522297047788687, 0.74215947158386420879381515431, 0.75113211551770793046289809912, 1.15076679071735357286544678873, 1.64688908564828840030864015019, 1.65792698040462913562445642436, 1.77674510500064850294721690132, 2.15076658063958127531879384002, 2.60690691233471318663579397865, 2.68020414429308571150446293530, 2.70598951697083816369289414866, 2.86189997625260014693917631706, 3.02182067640612861461793053916, 3.21356984783010834134914356705, 3.45353559249725831980869933792, 3.81054099366394431487385784116, 3.88210641351482324329811659325, 3.92874851580968128457985267202, 3.93609528969249748849854885758, 4.13554697444229815571300062099, 4.41286158516751917019911100796, 4.69977337626448632543905973978, 4.96458761714490718307784340808, 5.02670764644944520419100357850, 5.16158272560448824218734901506

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

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