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Properties

Label 12-2646e6-1.1-c1e6-0-4
Degree $12$
Conductor $3.432\times 10^{20}$
Sign $1$
Analytic cond. $8.89614\times 10^{7}$
Root an. cond. $4.59656$
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  − 6·2-s + 21·4-s − 5·5-s − 56·8-s + 30·10-s + 11-s + 2·13-s + 126·16-s − 4·17-s + 3·19-s − 105·20-s − 6·22-s + 7·23-s + 19·25-s − 12·26-s + 5·29-s − 28·31-s − 252·32-s + 24·34-s − 9·37-s − 18·38-s + 280·40-s − 12·41-s + 18·43-s + 21·44-s − 42·46-s − 6·47-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s − 2.23·5-s − 19.7·8-s + 9.48·10-s + 0.301·11-s + 0.554·13-s + 63/2·16-s − 0.970·17-s + 0.688·19-s − 23.4·20-s − 1.27·22-s + 1.45·23-s + 19/5·25-s − 2.35·26-s + 0.928·29-s − 5.02·31-s − 44.5·32-s + 4.11·34-s − 1.47·37-s − 2.91·38-s + 44.2·40-s − 1.87·41-s + 2.74·43-s + 3.16·44-s − 6.19·46-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(8.89614\times 10^{7}\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1593420500\)
\(L(\frac12)\) \(\approx\) \(0.1593420500\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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 \( 1 - 2 T - 32 T^{2} + 2 p T^{3} + 730 T^{4} - 230 T^{5} - 10729 T^{6} - 230 p T^{7} + 730 p^{2} T^{8} + 2 p^{4} T^{9} - 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 6020 p T^{7} - 185 p^{2} T^{8} + 371 p^{3} T^{9} - 30 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 14 T + 138 T^{2} + 841 T^{3} + 138 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 24546 p T^{7} + 7470 p^{2} T^{8} - 78 p^{3} T^{9} - 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 3 T + 117 T^{2} + 309 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 4 T + 76 T^{2} + 11 p T^{3} + 76 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 4 T + 48 T^{2} + 229 T^{3} + 48 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 32420 p T^{7} + 29404 p^{2} T^{8} + 518 p^{3} T^{9} - 180 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 28 T + 257 T^{2} - 2820 T^{3} + 59506 T^{4} - 545924 T^{5} + 3126001 T^{6} - 545924 p T^{7} + 59506 p^{2} T^{8} - 2820 p^{3} T^{9} + 257 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.65817091737084468178588731235, −4.48304715342769333629171383844, −4.22511282418307819920901091571, −4.01935433216376137538858218889, −3.67983098964066460589305440504, −3.62850273154343820887494115972, −3.59574182867722121551965956787, −3.57519086952350046865901301285, −3.33063009648671581403321973778, −3.12210553020404409762325861954, −3.12189739730348471685493445591, −2.69847925047286153055131521893, −2.60999375501207505877382282741, −2.45913456788336638515630413006, −2.31046248997104811515043688157, −2.11172579110647583709121723174, −1.74708074377013536581365554582, −1.58633030915895088682540271049, −1.56558449266887427138845847381, −1.26882124729969113457086842577, −1.22209507071214667740458342306, −0.900432817667553542524570120713, −0.47222047080707116243635470008, −0.30305154458335416861059542032, −0.29422409385035194604713376337, 0.29422409385035194604713376337, 0.30305154458335416861059542032, 0.47222047080707116243635470008, 0.900432817667553542524570120713, 1.22209507071214667740458342306, 1.26882124729969113457086842577, 1.56558449266887427138845847381, 1.58633030915895088682540271049, 1.74708074377013536581365554582, 2.11172579110647583709121723174, 2.31046248997104811515043688157, 2.45913456788336638515630413006, 2.60999375501207505877382282741, 2.69847925047286153055131521893, 3.12189739730348471685493445591, 3.12210553020404409762325861954, 3.33063009648671581403321973778, 3.57519086952350046865901301285, 3.59574182867722121551965956787, 3.62850273154343820887494115972, 3.67983098964066460589305440504, 4.01935433216376137538858218889, 4.22511282418307819920901091571, 4.48304715342769333629171383844, 4.65817091737084468178588731235

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

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