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Properties

Label 12-882e6-1.1-c1e6-0-5
Degree $12$
Conductor $4.708\times 10^{17}$
Sign $1$
Analytic cond. $122032.$
Root an. cond. $2.65382$
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 − 2·3-s + 3·4-s + 2·5-s − 6·6-s − 2·8-s + 6·10-s + 2·11-s − 6·12-s − 8·13-s − 4·15-s − 9·16-s + 4·17-s + 3·19-s + 6·20-s + 6·22-s + 14·23-s + 4·24-s − 15·25-s − 24·26-s + 5·27-s − 5·29-s − 12·30-s − 20·31-s − 9·32-s − 4·33-s + 12·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 2.44·6-s − 0.707·8-s + 1.89·10-s + 0.603·11-s − 1.73·12-s − 2.21·13-s − 1.03·15-s − 9/4·16-s + 0.970·17-s + 0.688·19-s + 1.34·20-s + 1.27·22-s + 2.91·23-s + 0.816·24-s − 3·25-s − 4.70·26-s + 0.962·27-s − 0.928·29-s − 2.19·30-s − 3.59·31-s − 1.59·32-s − 0.696·33-s + 2.05·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(122032.\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.708144157\)
\(L(\frac12)\) \(\approx\) \(2.708144157\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + 4 T^{2} + p T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good5 \( ( 1 - T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - T + 27 T^{2} - 25 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 8 T + 24 T^{2} + 42 T^{3} - 32 T^{4} - 1408 T^{5} - 7901 T^{6} - 1408 p T^{7} - 32 p^{2} T^{8} + 42 p^{3} T^{9} + 24 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 - 7 T + 81 T^{2} - 325 T^{3} + 81 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 3418 p T^{7} + 197 p^{2} T^{8} + 251 p^{3} T^{9} + 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
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 \( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 13564 p T^{7} + 18788 p^{2} T^{8} + 2 p^{3} T^{9} - 182 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 28 T + 281 T^{2} + 2724 T^{3} + 45178 T^{4} + 388196 T^{5} + 2169217 T^{6} + 388196 p T^{7} + 45178 p^{2} T^{8} + 2724 p^{3} T^{9} + 281 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.35354772291003960569278005938, −5.33008030029587114905917861928, −5.18993893585747778876285429393, −4.95078751094005910501521419290, −4.92406378904736415913657689935, −4.62514239265376033564564081336, −4.37771959156437190318007914439, −4.26082429886164228258604970531, −3.93041284134526325081779523297, −3.86439888202851027259706602745, −3.81064746550542420797907432895, −3.62640808473415162299507896563, −3.57886166778538079775157228155, −3.09536346883604719817534817938, −2.98124494861227508944443258946, −2.60515980569640017113562494682, −2.59332647475873462420720403912, −2.48828204563254151578096868014, −2.35122072651991706054154157876, −1.78828698966792937214516640034, −1.60982015257796281115018729157, −1.35123332733450804489928228884, −1.26516580058419491578603581842, −0.37829664396530860237396457369, −0.35676202302856068109599730474, 0.35676202302856068109599730474, 0.37829664396530860237396457369, 1.26516580058419491578603581842, 1.35123332733450804489928228884, 1.60982015257796281115018729157, 1.78828698966792937214516640034, 2.35122072651991706054154157876, 2.48828204563254151578096868014, 2.59332647475873462420720403912, 2.60515980569640017113562494682, 2.98124494861227508944443258946, 3.09536346883604719817534817938, 3.57886166778538079775157228155, 3.62640808473415162299507896563, 3.81064746550542420797907432895, 3.86439888202851027259706602745, 3.93041284134526325081779523297, 4.26082429886164228258604970531, 4.37771959156437190318007914439, 4.62514239265376033564564081336, 4.92406378904736415913657689935, 4.95078751094005910501521419290, 5.18993893585747778876285429393, 5.33008030029587114905917861928, 5.35354772291003960569278005938

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

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