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Properties

Label 12-4030e6-1.1-c1e6-0-4
Degree $12$
Conductor $4.284\times 10^{21}$
Sign $1$
Analytic cond. $1.11043\times 10^{9}$
Root an. cond. $5.67271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 3·3-s + 21·4-s + 6·5-s − 18·6-s − 10·7-s + 56·8-s − 5·9-s + 36·10-s − 10·11-s − 63·12-s − 6·13-s − 60·14-s − 18·15-s + 126·16-s − 4·17-s − 30·18-s − 7·19-s + 126·20-s + 30·21-s − 60·22-s − 3·23-s − 168·24-s + 21·25-s − 36·26-s + 26·27-s − 210·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 1.73·3-s + 21/2·4-s + 2.68·5-s − 7.34·6-s − 3.77·7-s + 19.7·8-s − 5/3·9-s + 11.3·10-s − 3.01·11-s − 18.1·12-s − 1.66·13-s − 16.0·14-s − 4.64·15-s + 63/2·16-s − 0.970·17-s − 7.07·18-s − 1.60·19-s + 28.1·20-s + 6.54·21-s − 12.7·22-s − 0.625·23-s − 34.2·24-s + 21/5·25-s − 7.06·26-s + 5.00·27-s − 39.6·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(1.11043\times 10^{9}\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
13 \( ( 1 + T )^{6} \)
31 \( ( 1 + T )^{6} \)
good3 \( 1 + p T + 14 T^{2} + 31 T^{3} + 89 T^{4} + 158 T^{5} + 341 T^{6} + 158 p T^{7} + 89 p^{2} T^{8} + 31 p^{3} T^{9} + 14 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + 10 T + 72 T^{2} + 368 T^{3} + 221 p T^{4} + 5263 T^{5} + 15297 T^{6} + 5263 p T^{7} + 221 p^{3} T^{8} + 368 p^{3} T^{9} + 72 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 + 5 T + 3 p T^{2} + 101 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 4 T + 29 T^{2} - 38 T^{3} + 22 p T^{4} + 29 p T^{5} + 17001 T^{6} + 29 p^{2} T^{7} + 22 p^{3} T^{8} - 38 p^{3} T^{9} + 29 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 7 T + 33 T^{2} + 79 T^{3} + 202 T^{4} + 1002 T^{5} + 7095 T^{6} + 1002 p T^{7} + 202 p^{2} T^{8} + 79 p^{3} T^{9} + 33 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 32 T^{2} + 159 T^{3} + 1170 T^{4} + 4179 T^{5} + 24093 T^{6} + 4179 p T^{7} + 1170 p^{2} T^{8} + 159 p^{3} T^{9} + 32 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 97 T^{2} - 36 T^{3} + 2004 T^{4} - 21899 T^{5} + 1045 T^{6} - 21899 p T^{7} + 2004 p^{2} T^{8} - 36 p^{3} T^{9} + 97 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 2 T + 165 T^{2} + 290 T^{3} + 12601 T^{4} + 19490 T^{5} + 582451 T^{6} + 19490 p T^{7} + 12601 p^{2} T^{8} + 290 p^{3} T^{9} + 165 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 10 T + 221 T^{2} + 1985 T^{3} + 21127 T^{4} + 160172 T^{5} + 1127279 T^{6} + 160172 p T^{7} + 21127 p^{2} T^{8} + 1985 p^{3} T^{9} + 221 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 181 T^{2} + 1253 T^{3} + 228 p T^{4} + 62928 T^{5} + 485135 T^{6} + 62928 p T^{7} + 228 p^{3} T^{8} + 1253 p^{3} T^{9} + 181 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 24 T + 382 T^{2} + 4834 T^{3} + 49347 T^{4} + 425085 T^{5} + 3155373 T^{6} + 425085 p T^{7} + 49347 p^{2} T^{8} + 4834 p^{3} T^{9} + 382 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 2 T + 211 T^{2} - 655 T^{3} + 22577 T^{4} - 63336 T^{5} + 1519581 T^{6} - 63336 p T^{7} + 22577 p^{2} T^{8} - 655 p^{3} T^{9} + 211 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 25 T + 330 T^{2} + 2327 T^{3} + 5858 T^{4} - 78727 T^{5} - 991403 T^{6} - 78727 p T^{7} + 5858 p^{2} T^{8} + 2327 p^{3} T^{9} + 330 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T + 259 T^{2} - 2339 T^{3} + 32405 T^{4} - 262671 T^{5} + 2466269 T^{6} - 262671 p T^{7} + 32405 p^{2} T^{8} - 2339 p^{3} T^{9} + 259 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 26 T + 526 T^{2} + 7488 T^{3} + 92199 T^{4} + 936089 T^{5} + 8347577 T^{6} + 936089 p T^{7} + 92199 p^{2} T^{8} + 7488 p^{3} T^{9} + 526 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 38 T + 11 p T^{2} + 10902 T^{3} + 115626 T^{4} + 1032326 T^{5} + 8693125 T^{6} + 1032326 p T^{7} + 115626 p^{2} T^{8} + 10902 p^{3} T^{9} + 11 p^{5} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + T + 131 T^{2} + 837 T^{3} + 13241 T^{4} + 57729 T^{5} + 1319959 T^{6} + 57729 p T^{7} + 13241 p^{2} T^{8} + 837 p^{3} T^{9} + 131 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 323 T^{2} - 264 T^{3} + 50707 T^{4} - 46006 T^{5} + 4980223 T^{6} - 46006 p T^{7} + 50707 p^{2} T^{8} - 264 p^{3} T^{9} + 323 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 18 T + 388 T^{2} + 4492 T^{3} + 51057 T^{4} + 469845 T^{5} + 4261607 T^{6} + 469845 p T^{7} + 51057 p^{2} T^{8} + 4492 p^{3} T^{9} + 388 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 4 T + 317 T^{2} - 946 T^{3} + 47566 T^{4} - 122809 T^{5} + 4881907 T^{6} - 122809 p T^{7} + 47566 p^{2} T^{8} - 946 p^{3} T^{9} + 317 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 21 T + 578 T^{2} + 7844 T^{3} + 126952 T^{4} + 1289787 T^{5} + 15612869 T^{6} + 1289787 p T^{7} + 126952 p^{2} T^{8} + 7844 p^{3} T^{9} + 578 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.92803603241101293494029580044, −4.76335819656839770594136562883, −4.55719930692309470957792911354, −4.44581292051944433179377149970, −4.44069473784300727360716472616, −4.15175635923279388379285232489, −3.99120210024961262411455580506, −3.52275336000762671731992147010, −3.43161159861942601218448934579, −3.41395779453875954296860723131, −3.32032068751174233136172470724, −3.28597854696934119982049556220, −3.18146963348243753330067269368, −2.79157255272415169651986944533, −2.78649906347016164662345601203, −2.65442331886384740805571900880, −2.56666051846758708496482413492, −2.51568100579321318622967412405, −2.41684411003929761603007223055, −2.15849444432487069918699123342, −1.84900083085015256002188108955, −1.65159581090736172596559456643, −1.60323339548256860405700365381, −1.44096026793157959599940722278, −1.25945716079953330796388844086, 0, 0, 0, 0, 0, 0, 1.25945716079953330796388844086, 1.44096026793157959599940722278, 1.60323339548256860405700365381, 1.65159581090736172596559456643, 1.84900083085015256002188108955, 2.15849444432487069918699123342, 2.41684411003929761603007223055, 2.51568100579321318622967412405, 2.56666051846758708496482413492, 2.65442331886384740805571900880, 2.78649906347016164662345601203, 2.79157255272415169651986944533, 3.18146963348243753330067269368, 3.28597854696934119982049556220, 3.32032068751174233136172470724, 3.41395779453875954296860723131, 3.43161159861942601218448934579, 3.52275336000762671731992147010, 3.99120210024961262411455580506, 4.15175635923279388379285232489, 4.44069473784300727360716472616, 4.44581292051944433179377149970, 4.55719930692309470957792911354, 4.76335819656839770594136562883, 4.92803603241101293494029580044

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

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