Properties

Label 12-2646e6-1.1-c1e6-0-3
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·4-s − 5·5-s − 2·8-s − 15·10-s + 11-s + 2·13-s − 9·16-s + 8·17-s − 6·19-s − 15·20-s + 3·22-s + 7·23-s + 19·25-s + 6·26-s + 5·29-s + 14·31-s − 9·32-s + 24·34-s + 18·37-s − 18·38-s + 10·40-s − 12·41-s + 18·43-s + 3·44-s + 21·46-s + 3·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 2.23·5-s − 0.707·8-s − 4.74·10-s + 0.301·11-s + 0.554·13-s − 9/4·16-s + 1.94·17-s − 1.37·19-s − 3.35·20-s + 0.639·22-s + 1.45·23-s + 19/5·25-s + 1.17·26-s + 0.928·29-s + 2.51·31-s − 1.59·32-s + 4.11·34-s + 2.95·37-s − 2.91·38-s + 1.58·40-s − 1.87·41-s + 2.74·43-s + 0.452·44-s + 3.09·46-s + 0.437·47-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

\(L(1)\) \(\approx\) \(10.03854915\)
\(L(\frac12)\) \(\approx\) \(10.03854915\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
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 \( ( 1 - 4 T + 7 T^{2} + 32 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 - 14 T + 58 T^{2} - 250 T^{3} + 2992 T^{4} - 9728 T^{5} - 11857 T^{6} - 9728 p T^{7} + 2992 p^{2} T^{8} - 250 p^{3} T^{9} + 58 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 9 T + 102 T^{2} - 593 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 9786 p T^{7} + 7263 p^{2} T^{8} + 267 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 9 T + 117 T^{2} - 963 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 464 p T^{5} + 7381 p T^{6} - 464 p^{2} T^{7} - 1304 p^{2} T^{8} + 994 p^{3} T^{9} - 60 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 32 T^{2} + 650 T^{3} + 292 T^{4} - 19532 T^{5} + 306323 T^{6} - 19532 p T^{7} + 292 p^{2} T^{8} + 650 p^{3} T^{9} - 32 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 + 25 T + 371 T^{2} + 3601 T^{3} + 371 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 - 9 T + 261 T^{2} - 1539 T^{3} + 261 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−4.49742349757444043070551312955, −4.41748851118740803216726022807, −4.40838995269031300970047171968, −4.27275974145317173924681500166, −4.07292421425174863511316567206, −3.97178420989055702517941403302, −3.64716044219094534818172631278, −3.56490555905005668295026486914, −3.42336300703873058217754732105, −3.40950710913873061793348099347, −3.10053816921755083762290948468, −3.08918064842860420899952163914, −2.79366350042823939024729892943, −2.73266732570780981515795935139, −2.57273450342307717096804763365, −2.48165559583459091655165003788, −2.19617762459763050916397031598, −2.08500063811174996808168102549, −1.47249698233592522459440435490, −1.38293477073359602657768335207, −1.11866734997559961697648318597, −1.00357531323638112860756686465, −0.71740705587828472688368475774, −0.69207241045606448748774346327, −0.24522628577142820703704552025, 0.24522628577142820703704552025, 0.69207241045606448748774346327, 0.71740705587828472688368475774, 1.00357531323638112860756686465, 1.11866734997559961697648318597, 1.38293477073359602657768335207, 1.47249698233592522459440435490, 2.08500063811174996808168102549, 2.19617762459763050916397031598, 2.48165559583459091655165003788, 2.57273450342307717096804763365, 2.73266732570780981515795935139, 2.79366350042823939024729892943, 3.08918064842860420899952163914, 3.10053816921755083762290948468, 3.40950710913873061793348099347, 3.42336300703873058217754732105, 3.56490555905005668295026486914, 3.64716044219094534818172631278, 3.97178420989055702517941403302, 4.07292421425174863511316567206, 4.27275974145317173924681500166, 4.40838995269031300970047171968, 4.41748851118740803216726022807, 4.49742349757444043070551312955

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

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