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Properties

Label 12-8470e6-1.1-c1e6-0-5
Degree $12$
Conductor $3.692\times 10^{23}$
Sign $1$
Analytic cond. $9.57112\times 10^{10}$
Root an. cond. $8.22394$
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 + 4·3-s + 21·4-s + 6·5-s + 24·6-s + 6·7-s + 56·8-s + 36·10-s + 84·12-s + 36·14-s + 24·15-s + 126·16-s + 2·17-s + 126·20-s + 24·21-s + 4·23-s + 224·24-s + 21·25-s − 20·27-s + 126·28-s + 8·29-s + 144·30-s + 4·31-s + 252·32-s + 12·34-s + 36·35-s + 4·37-s + ⋯
L(s)  = 1  + 4.24·2-s + 2.30·3-s + 21/2·4-s + 2.68·5-s + 9.79·6-s + 2.26·7-s + 19.7·8-s + 11.3·10-s + 24.2·12-s + 9.62·14-s + 6.19·15-s + 63/2·16-s + 0.485·17-s + 28.1·20-s + 5.23·21-s + 0.834·23-s + 45.7·24-s + 21/5·25-s − 3.84·27-s + 23.8·28-s + 1.48·29-s + 26.2·30-s + 0.718·31-s + 44.5·32-s + 2.05·34-s + 6.08·35-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.57112\times 10^{10}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6205.485006\)
\(L(\frac12)\) \(\approx\) \(6205.485006\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
5 \( ( 1 - T )^{6} \)
7 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( ( 1 - 2 T + 2 p T^{2} - 10 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 36 T^{2} + 792 T^{4} + 12710 T^{6} + 792 p^{2} T^{8} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - T + 27 T^{2} - 72 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 + 73 T^{2} + 80 T^{3} + 2582 T^{4} + 3616 T^{5} + 59837 T^{6} + 3616 p T^{7} + 2582 p^{2} T^{8} + 80 p^{3} T^{9} + 73 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 4 T + 88 T^{2} - 192 T^{3} + 3444 T^{4} - 4948 T^{5} + 92586 T^{6} - 4948 p T^{7} + 3444 p^{2} T^{8} - 192 p^{3} T^{9} + 88 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 8 T + 138 T^{2} - 824 T^{3} + 8087 T^{4} - 38688 T^{5} + 285964 T^{6} - 38688 p T^{7} + 8087 p^{2} T^{8} - 824 p^{3} T^{9} + 138 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 2 T + 87 T^{2} - 116 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 4 T + 166 T^{2} - 628 T^{3} + 12647 T^{4} - 42712 T^{5} + 582164 T^{6} - 42712 p T^{7} + 12647 p^{2} T^{8} - 628 p^{3} T^{9} + 166 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 12 T + 6 p T^{2} - 2140 T^{3} + 24339 T^{4} - 161928 T^{5} + 1306556 T^{6} - 161928 p T^{7} + 24339 p^{2} T^{8} - 2140 p^{3} T^{9} + 6 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 6 T + 199 T^{2} + 722 T^{3} + 16199 T^{4} + 37132 T^{5} + 817238 T^{6} + 37132 p T^{7} + 16199 p^{2} T^{8} + 722 p^{3} T^{9} + 199 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 16 T + 222 T^{2} - 2272 T^{3} + 21731 T^{4} - 167088 T^{5} + 1242700 T^{6} - 167088 p T^{7} + 21731 p^{2} T^{8} - 2272 p^{3} T^{9} + 222 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 12 T + 327 T^{2} - 2932 T^{3} + 43383 T^{4} - 297504 T^{5} + 3059702 T^{6} - 297504 p T^{7} + 43383 p^{2} T^{8} - 2932 p^{3} T^{9} + 327 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 22 T + 465 T^{2} - 5902 T^{3} + 71450 T^{4} - 10866 p T^{5} + 5584849 T^{6} - 10866 p^{2} T^{7} + 71450 p^{2} T^{8} - 5902 p^{3} T^{9} + 465 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T + 191 T^{2} + 1268 T^{3} + 17579 T^{4} + 154296 T^{5} + 1171290 T^{6} + 154296 p T^{7} + 17579 p^{2} T^{8} + 1268 p^{3} T^{9} + 191 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 20 T + 419 T^{2} - 4876 T^{3} + 60863 T^{4} - 524904 T^{5} + 5041134 T^{6} - 524904 p T^{7} + 60863 p^{2} T^{8} - 4876 p^{3} T^{9} + 419 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 14 T + 179 T^{2} - 1730 T^{3} + 17995 T^{4} - 129156 T^{5} + 1064258 T^{6} - 129156 p T^{7} + 17995 p^{2} T^{8} - 1730 p^{3} T^{9} + 179 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 18 T + 463 T^{2} - 5834 T^{3} + 85631 T^{4} - 804076 T^{5} + 8341718 T^{6} - 804076 p T^{7} + 85631 p^{2} T^{8} - 5834 p^{3} T^{9} + 463 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 32 T + 693 T^{2} - 10528 T^{3} + 129578 T^{4} - 1355040 T^{5} + 12612349 T^{6} - 1355040 p T^{7} + 129578 p^{2} T^{8} - 10528 p^{3} T^{9} + 693 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 16 T + 368 T^{2} + 4960 T^{3} + 66256 T^{4} + 697296 T^{5} + 7046630 T^{6} + 697296 p T^{7} + 66256 p^{2} T^{8} + 4960 p^{3} T^{9} + 368 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 4 T + 110 T^{2} + 572 T^{3} + 8323 T^{4} - 25800 T^{5} + 1678316 T^{6} - 25800 p T^{7} + 8323 p^{2} T^{8} + 572 p^{3} T^{9} + 110 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 331 T^{2} - 2128 T^{3} + 58235 T^{4} - 375292 T^{5} + 6941090 T^{6} - 375292 p T^{7} + 58235 p^{2} T^{8} - 2128 p^{3} T^{9} + 331 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.11547546350474263545312055601, −3.64693151222343077159291431784, −3.48373270895975164889535566770, −3.47136102177913093919227738784, −3.46997565712715089435385083093, −3.42741281017341878587908504774, −3.29926161023905978901976979662, −2.98300664370792009045225209135, −2.80708026547922481827297384406, −2.65437379809585017886897965955, −2.60529867143654073414288384609, −2.56077824661636780836285982655, −2.39046225461683178300687544837, −2.21564991979225514355121374008, −2.14864583348738465020176473974, −2.14593640047379141668598507459, −2.14410774574026698816559044689, −1.82646060980203461392398379348, −1.74142416255422947762213437232, −1.21288998203796523731794607529, −1.20074143329364804525213671086, −1.01499738223630044462905441049, −0.851192115674593447286541213651, −0.821235043728869545470673534759, −0.60584279560376570186684891717, 0.60584279560376570186684891717, 0.821235043728869545470673534759, 0.851192115674593447286541213651, 1.01499738223630044462905441049, 1.20074143329364804525213671086, 1.21288998203796523731794607529, 1.74142416255422947762213437232, 1.82646060980203461392398379348, 2.14410774574026698816559044689, 2.14593640047379141668598507459, 2.14864583348738465020176473974, 2.21564991979225514355121374008, 2.39046225461683178300687544837, 2.56077824661636780836285982655, 2.60529867143654073414288384609, 2.65437379809585017886897965955, 2.80708026547922481827297384406, 2.98300664370792009045225209135, 3.29926161023905978901976979662, 3.42741281017341878587908504774, 3.46997565712715089435385083093, 3.47136102177913093919227738784, 3.48373270895975164889535566770, 3.64693151222343077159291431784, 4.11547546350474263545312055601

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

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