Properties

Label 12-912e6-1.1-c1e6-0-9
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  + 3·5-s + 3·7-s + 9·13-s − 3·17-s − 27·23-s − 3·25-s − 27-s − 3·29-s + 15·31-s + 9·35-s − 6·37-s − 15·41-s − 3·43-s − 15·47-s + 3·49-s + 6·53-s + 27·59-s − 15·61-s + 27·65-s + 3·67-s − 3·71-s + 12·73-s − 27·79-s − 3·83-s − 9·85-s + 42·89-s + 27·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s + 2.49·13-s − 0.727·17-s − 5.62·23-s − 3/5·25-s − 0.192·27-s − 0.557·29-s + 2.69·31-s + 1.52·35-s − 0.986·37-s − 2.34·41-s − 0.457·43-s − 2.18·47-s + 3/7·49-s + 0.824·53-s + 3.51·59-s − 1.92·61-s + 3.34·65-s + 0.366·67-s − 0.356·71-s + 1.40·73-s − 3.03·79-s − 0.329·83-s − 0.976·85-s + 4.45·89-s + 2.83·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.420800587\)
\(L(\frac12)\) \(\approx\) \(4.420800587\)
\(L(\frac{3}{2})\) 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 \( 1 + T^{3} + T^{6} \)
19 \( 1 - 107 T^{3} + p^{3} T^{6} \)
good5 \( 1 - 3 T + 12 T^{2} - 16 T^{3} + 3 p^{2} T^{4} - 153 T^{5} + 581 T^{6} - 153 p T^{7} + 3 p^{4} T^{8} - 16 p^{3} T^{9} + 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T + 6 T^{2} - 39 T^{3} + 33 T^{4} + 24 T^{5} + 407 T^{6} + 24 p T^{7} + 33 p^{2} T^{8} - 39 p^{3} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 12 T^{2} - 74 T^{3} + 12 T^{4} + 444 T^{5} + 2447 T^{6} + 444 p T^{7} + 12 p^{2} T^{8} - 74 p^{3} T^{9} - 12 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 106 T^{3} + 27 T^{4} + 1161 T^{5} - 4731 T^{6} + 1161 p T^{7} + 27 p^{2} T^{8} - 106 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T + 18 T^{2} + 126 T^{3} + 621 T^{4} + 2217 T^{5} + 10549 T^{6} + 2217 p T^{7} + 621 p^{2} T^{8} + 126 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 27 T + 378 T^{2} + 3694 T^{3} + 28215 T^{4} + 177039 T^{5} + 927971 T^{6} + 177039 p T^{7} + 28215 p^{2} T^{8} + 3694 p^{3} T^{9} + 378 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 12 T^{2} - 128 T^{3} + 147 T^{4} - 3231 T^{5} - 1507 T^{6} - 3231 p T^{7} + 147 p^{2} T^{8} - 128 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 15 T + 60 T^{2} - 397 T^{3} + 6525 T^{4} - 32400 T^{5} + 78903 T^{6} - 32400 p T^{7} + 6525 p^{2} T^{8} - 397 p^{3} T^{9} + 60 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 66 T^{2} + 239 T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 87 T^{2} - 53 T^{3} - 2874 T^{4} - 20052 T^{5} - 101167 T^{6} - 20052 p T^{7} - 2874 p^{2} T^{8} - 53 p^{3} T^{9} + 87 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T - 3 T^{2} + 107 T^{3} - 1206 T^{4} - 3258 T^{5} + 55089 T^{6} - 3258 p T^{7} - 1206 p^{2} T^{8} + 107 p^{3} T^{9} - 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 15 T + 51 T^{2} - 503 T^{3} - 4224 T^{4} + 24030 T^{5} + 424469 T^{6} + 24030 p T^{7} - 4224 p^{2} T^{8} - 503 p^{3} T^{9} + 51 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 30 T^{2} - 226 T^{3} + 2688 T^{4} - 11340 T^{5} + 185231 T^{6} - 11340 p T^{7} + 2688 p^{2} T^{8} - 226 p^{3} T^{9} - 30 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 27 T + 198 T^{2} + 1134 T^{3} - 21663 T^{4} + 27117 T^{5} + 768025 T^{6} + 27117 p T^{7} - 21663 p^{2} T^{8} + 1134 p^{3} T^{9} + 198 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 15 T + 84 T^{2} + 32 T^{3} - 1638 T^{4} - 9657 T^{5} - 118173 T^{6} - 9657 p T^{7} - 1638 p^{2} T^{8} + 32 p^{3} T^{9} + 84 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T + 72 T^{2} - 30 T^{3} + 4257 T^{4} + 9465 T^{5} + 51479 T^{6} + 9465 p T^{7} + 4257 p^{2} T^{8} - 30 p^{3} T^{9} + 72 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 3 T - 87 T^{2} + 187 T^{3} - 2094 T^{4} - 26118 T^{5} + 499145 T^{6} - 26118 p T^{7} - 2094 p^{2} T^{8} + 187 p^{3} T^{9} - 87 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 12 T + 54 T^{2} - 795 T^{3} - 135 T^{4} + 40083 T^{5} + 66905 T^{6} + 40083 p T^{7} - 135 p^{2} T^{8} - 795 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 351 T^{2} + 3337 T^{3} + 19872 T^{4} + 49734 T^{5} + 58989 T^{6} + 49734 p T^{7} + 19872 p^{2} T^{8} + 3337 p^{3} T^{9} + 351 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 3 T - 204 T^{2} - 459 T^{3} + 26043 T^{4} + 31530 T^{5} - 2382653 T^{6} + 31530 p T^{7} + 26043 p^{2} T^{8} - 459 p^{3} T^{9} - 204 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 42 T + 756 T^{2} - 7371 T^{3} + 29799 T^{4} + 258951 T^{5} - 4921811 T^{6} + 258951 p T^{7} + 29799 p^{2} T^{8} - 7371 p^{3} T^{9} + 756 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 171 T^{2} - 713 T^{3} - 4239 T^{4} + 76491 T^{5} - 635802 T^{6} + 76491 p T^{7} - 4239 p^{2} T^{8} - 713 p^{3} T^{9} + 171 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.26339731004637606498042926814, −5.23448674314607879384026985247, −5.23443945525094831282179516213, −5.00335477013949974122329356669, −4.52941278077148615100382150434, −4.47446856265032892073162926554, −4.38866201462903379860327832906, −4.20976033835653443962388958444, −3.97838737488322521595321320866, −3.80250240178029009434576367535, −3.75056119113168975113966378756, −3.48860813624675571914527641771, −3.46198727058996194710240467539, −3.09911557520587182029318220371, −2.90602185090911719357834615165, −2.56524490893463170412062938222, −2.18871816553686500591652462504, −2.15532066579000429664137820055, −2.12577680163078965239831157024, −1.82632904728196491194209253247, −1.51641279786186936437916750209, −1.43980836266742677590371950763, −1.41458045871151046441376280029, −0.55231471161788936492440829642, −0.37637696916551082606209091099, 0.37637696916551082606209091099, 0.55231471161788936492440829642, 1.41458045871151046441376280029, 1.43980836266742677590371950763, 1.51641279786186936437916750209, 1.82632904728196491194209253247, 2.12577680163078965239831157024, 2.15532066579000429664137820055, 2.18871816553686500591652462504, 2.56524490893463170412062938222, 2.90602185090911719357834615165, 3.09911557520587182029318220371, 3.46198727058996194710240467539, 3.48860813624675571914527641771, 3.75056119113168975113966378756, 3.80250240178029009434576367535, 3.97838737488322521595321320866, 4.20976033835653443962388958444, 4.38866201462903379860327832906, 4.47446856265032892073162926554, 4.52941278077148615100382150434, 5.00335477013949974122329356669, 5.23443945525094831282179516213, 5.23448674314607879384026985247, 5.26339731004637606498042926814

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

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