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Properties

Label 12-1134e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.127\times 10^{18}$
Sign $1$
Analytic cond. $551240.$
Root an. cond. $3.00915$
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  − 3·2-s + 3·4-s − 5-s + 2·7-s + 2·8-s + 3·10-s + 11-s − 16·13-s − 6·14-s − 9·16-s + 4·17-s − 3·19-s − 3·20-s − 3·22-s + 7·23-s + 9·25-s + 48·26-s + 6·28-s − 10·29-s + 20·31-s + 9·32-s − 12·34-s − 2·35-s + 3·37-s + 9·38-s − 2·40-s + 12·43-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 0.707·8-s + 0.948·10-s + 0.301·11-s − 4.43·13-s − 1.60·14-s − 9/4·16-s + 0.970·17-s − 0.688·19-s − 0.670·20-s − 0.639·22-s + 1.45·23-s + 9/5·25-s + 9.41·26-s + 1.13·28-s − 1.85·29-s + 3.59·31-s + 1.59·32-s − 2.05·34-s − 0.338·35-s + 0.493·37-s + 1.45·38-s − 0.316·40-s + 1.82·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{24} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(551240.\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5460866819\)
\(L(\frac12)\) \(\approx\) \(0.5460866819\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 52 p T^{7} + 23 p^{2} T^{8} - 17 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T - 26 T^{2} + 23 T^{3} + 37 p T^{4} - 202 T^{5} - 4853 T^{6} - 202 p T^{7} + 37 p^{3} T^{8} + 23 p^{3} T^{9} - 26 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 8 T + 40 T^{2} + 139 T^{3} + 40 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 4 T - 23 T^{2} + 4 p T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 220 p T^{7} + 410 p^{2} T^{8} + 4 p^{4} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 3946 p T^{7} + 2423 p^{2} T^{8} + 83 p^{3} T^{9} - 32 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 5 T + 21 T^{2} - 73 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
41 \( ( 1 + 90 T^{2} - 9 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 6 T + 60 T^{2} - 389 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 60360 p T^{7} + 13635 p^{2} T^{8} + 33 p^{3} T^{9} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 35126 p T^{7} + 11666 p^{2} T^{8} + 154 p^{3} T^{9} - 20 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 2 T + 186 T^{2} - 185 T^{3} + 186 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 73314 p T^{7} + 16101 p^{2} T^{8} + 1197 p^{3} T^{9} - 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 503 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.05314678469189144969487658657, −4.96576778432072844046420413285, −4.91762161926709045036768818197, −4.89972695051314991335802751009, −4.48882644984893474892041174544, −4.33750357077572306333313510682, −4.20975984926239227324322039834, −4.16115206835326002881985035987, −4.14180611614753255039484140957, −3.60700205210818196620808211148, −3.55450886361861631402998962478, −3.20104012483558732629234410214, −2.93836736713722294963511673337, −2.86524750312932327889793592580, −2.55406322785989776867499544300, −2.48221819954047393544328565061, −2.44964895616428726458668944538, −2.29968485884261114315892631214, −1.77996875609968237665201028803, −1.62538141777915267644778850861, −1.36749441425256368159136127830, −1.04573706535133229194665909459, −0.74704766647257163301545222496, −0.65279016507163005034804944902, −0.24859858381071957942222525630, 0.24859858381071957942222525630, 0.65279016507163005034804944902, 0.74704766647257163301545222496, 1.04573706535133229194665909459, 1.36749441425256368159136127830, 1.62538141777915267644778850861, 1.77996875609968237665201028803, 2.29968485884261114315892631214, 2.44964895616428726458668944538, 2.48221819954047393544328565061, 2.55406322785989776867499544300, 2.86524750312932327889793592580, 2.93836736713722294963511673337, 3.20104012483558732629234410214, 3.55450886361861631402998962478, 3.60700205210818196620808211148, 4.14180611614753255039484140957, 4.16115206835326002881985035987, 4.20975984926239227324322039834, 4.33750357077572306333313510682, 4.48882644984893474892041174544, 4.89972695051314991335802751009, 4.91762161926709045036768818197, 4.96576778432072844046420413285, 5.05314678469189144969487658657

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

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