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Properties

Label 12-6468e6-1.1-c1e6-0-1
Degree $12$
Conductor $7.322\times 10^{22}$
Sign $1$
Analytic cond. $1.89794\times 10^{10}$
Root an. cond. $7.18660$
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  + 6·3-s + 4·5-s + 21·9-s + 6·11-s + 8·13-s + 24·15-s + 8·17-s + 4·19-s + 8·23-s − 4·25-s + 56·27-s + 4·29-s + 8·31-s + 36·33-s + 4·37-s + 48·39-s + 4·41-s + 8·43-s + 84·45-s + 4·47-s + 48·51-s − 4·53-s + 24·55-s + 24·57-s + 16·59-s + 32·65-s − 8·67-s + ⋯
L(s)  = 1  + 3.46·3-s + 1.78·5-s + 7·9-s + 1.80·11-s + 2.21·13-s + 6.19·15-s + 1.94·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s + 10.7·27-s + 0.742·29-s + 1.43·31-s + 6.26·33-s + 0.657·37-s + 7.68·39-s + 0.624·41-s + 1.21·43-s + 12.5·45-s + 0.583·47-s + 6.72·51-s − 0.549·53-s + 3.23·55-s + 3.17·57-s + 2.08·59-s + 3.96·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 7^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 7^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{6} \cdot 7^{12} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(1.89794\times 10^{10}\)
Root analytic conductor: \(7.18660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{6} \cdot 7^{12} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(535.9320304\)
\(L(\frac12)\) \(\approx\) \(535.9320304\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
7 \( 1 \)
11 \( ( 1 - T )^{6} \)
good5 \( 1 - 4 T + 4 p T^{2} - 12 p T^{3} + 198 T^{4} - 496 T^{5} + 1248 T^{6} - 496 p T^{7} + 198 p^{2} T^{8} - 12 p^{4} T^{9} + 4 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 66 T^{2} - 376 T^{3} + 148 p T^{4} - 8040 T^{5} + 32124 T^{6} - 8040 p T^{7} + 148 p^{3} T^{8} - 376 p^{3} T^{9} + 66 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + 94 T^{2} - 520 T^{3} + 3551 T^{4} - 15024 T^{5} + 76284 T^{6} - 15024 p T^{7} + 3551 p^{2} T^{8} - 520 p^{3} T^{9} + 94 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 4 T + 72 T^{2} - 240 T^{3} + 2536 T^{4} - 7548 T^{5} + 58670 T^{6} - 7548 p T^{7} + 2536 p^{2} T^{8} - 240 p^{3} T^{9} + 72 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 8 T + 108 T^{2} - 576 T^{3} + 5043 T^{4} - 22168 T^{5} + 148320 T^{6} - 22168 p T^{7} + 5043 p^{2} T^{8} - 576 p^{3} T^{9} + 108 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 86 T^{2} - 276 T^{3} + 4014 T^{4} - 11264 T^{5} + 137722 T^{6} - 11264 p T^{7} + 4014 p^{2} T^{8} - 276 p^{3} T^{9} + 86 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 8 T + 116 T^{2} - 784 T^{3} + 7155 T^{4} - 39688 T^{5} + 267408 T^{6} - 39688 p T^{7} + 7155 p^{2} T^{8} - 784 p^{3} T^{9} + 116 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T + 100 T^{2} - 104 T^{3} + 3328 T^{4} + 14076 T^{5} + 71986 T^{6} + 14076 p T^{7} + 3328 p^{2} T^{8} - 104 p^{3} T^{9} + 100 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 140 T^{2} - 852 T^{3} + 9347 T^{4} - 71224 T^{5} + 429720 T^{6} - 71224 p T^{7} + 9347 p^{2} T^{8} - 852 p^{3} T^{9} + 140 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 132 T^{2} - 864 T^{3} + 6827 T^{4} - 43320 T^{5} + 251728 T^{6} - 43320 p T^{7} + 6827 p^{2} T^{8} - 864 p^{3} T^{9} + 132 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 4 T + 106 T^{2} - 292 T^{3} + 3846 T^{4} - 7752 T^{5} + 101430 T^{6} - 7752 p T^{7} + 3846 p^{2} T^{8} - 292 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 140 T^{2} + 196 T^{3} + 10931 T^{4} + 13496 T^{5} + 694864 T^{6} + 13496 p T^{7} + 10931 p^{2} T^{8} + 196 p^{3} T^{9} + 140 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 16 T + 290 T^{2} - 52 p T^{3} + 34150 T^{4} - 4732 p T^{5} + 2425814 T^{6} - 4732 p^{2} T^{7} + 34150 p^{2} T^{8} - 52 p^{4} T^{9} + 290 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 228 T^{2} + 240 T^{3} + 26239 T^{4} + 33744 T^{5} + 1961640 T^{6} + 33744 p T^{7} + 26239 p^{2} T^{8} + 240 p^{3} T^{9} + 228 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 + 8 T + 168 T^{2} + 616 T^{3} + 13144 T^{4} + 17544 T^{5} + 808806 T^{6} + 17544 p T^{7} + 13144 p^{2} T^{8} + 616 p^{3} T^{9} + 168 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T + 386 T^{2} - 3596 T^{3} + 63643 T^{4} - 469600 T^{5} + 5871780 T^{6} - 469600 p T^{7} + 63643 p^{2} T^{8} - 3596 p^{3} T^{9} + 386 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 20 T + 306 T^{2} - 3688 T^{3} + 42252 T^{4} - 427444 T^{5} + 4036644 T^{6} - 427444 p T^{7} + 42252 p^{2} T^{8} - 3688 p^{3} T^{9} + 306 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 386 T^{2} + 4208 T^{3} + 769 p T^{4} + 514912 T^{5} + 5816860 T^{6} + 514912 p T^{7} + 769 p^{3} T^{8} + 4208 p^{3} T^{9} + 386 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 8 T + 340 T^{2} - 1768 T^{3} + 47411 T^{4} - 161792 T^{5} + 4328480 T^{6} - 161792 p T^{7} + 47411 p^{2} T^{8} - 1768 p^{3} T^{9} + 340 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 4 T + 408 T^{2} - 732 T^{3} + 71311 T^{4} - 35008 T^{5} + 7639984 T^{6} - 35008 p T^{7} + 71311 p^{2} T^{8} - 732 p^{3} T^{9} + 408 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 24 T + 646 T^{2} - 9808 T^{3} + 157319 T^{4} - 1754280 T^{5} + 20277580 T^{6} - 1754280 p T^{7} + 157319 p^{2} T^{8} - 9808 p^{3} T^{9} + 646 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.02564567054369922793422649928, −3.86468248129809363123980852370, −3.68968692454333762273555706434, −3.65134812249550325025656647428, −3.58311847628360514659527847887, −3.33353931836150648966826889251, −3.30239422029313121888997155306, −2.95515622227149616533535290535, −2.94886597977161653894731124447, −2.82736453579942457731495684593, −2.80788466209697040264362141474, −2.62409025806409896053680122882, −2.57898034873120085987115711306, −2.02779840734128995042196727348, −1.94746528151501992983464106371, −1.87446274316742623249990177634, −1.84247614037552577558653908581, −1.83922227145074692953404982680, −1.64242746381365566425429601176, −1.26819877879712366601852696392, −1.00478758749422612332845433251, −0.953790839670870007041780758834, −0.792096507659614834428389172049, −0.73341828114195076372258181933, −0.72114708083924817207012587383, 0.72114708083924817207012587383, 0.73341828114195076372258181933, 0.792096507659614834428389172049, 0.953790839670870007041780758834, 1.00478758749422612332845433251, 1.26819877879712366601852696392, 1.64242746381365566425429601176, 1.83922227145074692953404982680, 1.84247614037552577558653908581, 1.87446274316742623249990177634, 1.94746528151501992983464106371, 2.02779840734128995042196727348, 2.57898034873120085987115711306, 2.62409025806409896053680122882, 2.80788466209697040264362141474, 2.82736453579942457731495684593, 2.94886597977161653894731124447, 2.95515622227149616533535290535, 3.30239422029313121888997155306, 3.33353931836150648966826889251, 3.58311847628360514659527847887, 3.65134812249550325025656647428, 3.68968692454333762273555706434, 3.86468248129809363123980852370, 4.02564567054369922793422649928

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

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