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Properties

Label 12-2646e6-1.1-c1e6-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s − 2·5-s + 2·8-s + 6·10-s − 2·11-s − 8·13-s − 9·16-s − 4·17-s + 3·19-s − 6·20-s + 6·22-s − 14·23-s − 15·25-s + 24·26-s + 5·29-s − 20·31-s + 9·32-s + 12·34-s + 3·37-s − 9·38-s − 4·40-s − 6·43-s − 6·44-s + 42·46-s − 9·47-s + 45·50-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.894·5-s + 0.707·8-s + 1.89·10-s − 0.603·11-s − 2.21·13-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 − 3·25-s + 4.70·26-s + 0.928·29-s − 3.59·31-s + 1.59·32-s + 2.05·34-s + 0.493·37-s − 1.45·38-s − 0.632·40-s − 0.914·43-s − 0.904·44-s + 6.19·46-s − 1.31·47-s + 6.36·50-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

\(L(1)\) \(\approx\) \(0.05937727559\)
\(L(\frac12)\) \(\approx\) \(0.05937727559\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T + 9 T^{2} + 13 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−4.71383326266261165764143884865, −4.43725337664072724268039868520, −4.26062710464139396347368206101, −4.02935023614271208106957541005, −4.00127049199699868644782249842, −3.94711289109007552228311363075, −3.83260677269417275106045533760, −3.49369834366745149090777647308, −3.32028113670895420689289483959, −3.17120428188905476849618366368, −3.12820886430964922037372299198, −2.81836439384658533318556635292, −2.71205462837721766423538016649, −2.42045039607319416795710394791, −2.22422819581699282909599387864, −2.14046664033809257741214564003, −1.88913144548316399530390403789, −1.64319936287207927053557051848, −1.63946970784245799611940640114, −1.51204809033762697135603639441, −1.48248403363810641020347091196, −0.68223168755965770565846557042, −0.37188822306373035435621985799, −0.24790360963850521169569613457, −0.20147753699830051876794276380, 0.20147753699830051876794276380, 0.24790360963850521169569613457, 0.37188822306373035435621985799, 0.68223168755965770565846557042, 1.48248403363810641020347091196, 1.51204809033762697135603639441, 1.63946970784245799611940640114, 1.64319936287207927053557051848, 1.88913144548316399530390403789, 2.14046664033809257741214564003, 2.22422819581699282909599387864, 2.42045039607319416795710394791, 2.71205462837721766423538016649, 2.81836439384658533318556635292, 3.12820886430964922037372299198, 3.17120428188905476849618366368, 3.32028113670895420689289483959, 3.49369834366745149090777647308, 3.83260677269417275106045533760, 3.94711289109007552228311363075, 4.00127049199699868644782249842, 4.02935023614271208106957541005, 4.26062710464139396347368206101, 4.43725337664072724268039868520, 4.71383326266261165764143884865

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

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