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Properties

Label 12-1008e6-1.1-c1e6-0-6
Degree $12$
Conductor $1.049\times 10^{18}$
Sign $1$
Analytic cond. $271910.$
Root an. cond. $2.83706$
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  − 2·3-s + 5-s − 2·7-s + 6·9-s + 11-s + 8·13-s − 2·15-s − 4·17-s + 3·19-s + 4·21-s + 7·23-s + 9·25-s − 7·27-s − 5·29-s + 40·31-s − 2·33-s − 2·35-s + 3·37-s − 16·39-s + 6·43-s + 6·45-s − 18·47-s + 8·49-s + 8·51-s + 15·53-s + 55-s − 6·57-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.755·7-s + 2·9-s + 0.301·11-s + 2.21·13-s − 0.516·15-s − 0.970·17-s + 0.688·19-s + 0.872·21-s + 1.45·23-s + 9/5·25-s − 1.34·27-s − 0.928·29-s + 7.18·31-s − 0.348·33-s − 0.338·35-s + 0.493·37-s − 2.56·39-s + 0.914·43-s + 0.894·45-s − 2.62·47-s + 8/7·49-s + 1.12·51-s + 2.06·53-s + 0.134·55-s − 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(271910.\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.914780137\)
\(L(\frac12)\) \(\approx\) \(4.914780137\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T - 2 T^{2} - p^{2} T^{3} - 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 + 2 T - 4 T^{2} - 31 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
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 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 3946 p T^{7} + 2423 p^{2} T^{8} + 83 p^{3} T^{9} - 32 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 3418 p T^{7} + 197 p^{2} T^{8} + 251 p^{3} T^{9} + 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 20 T + 214 T^{2} - 1441 T^{3} + 214 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 6 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( ( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 216 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - T + 89 T^{2} + 77 T^{3} + 89 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 5 T + 163 T^{2} - 469 T^{3} + 163 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−5.35813398189847867956567741615, −5.01478102127000862832057924351, −4.89397526299315625053911525482, −4.74233028439509593182627061415, −4.65289890291381468538172867432, −4.53735514978530019890637063504, −4.42408101901510414349463888895, −4.23275347176796323784113961632, −3.89020791889088343854179947962, −3.80587243408054002686252434227, −3.72145895952480140282974803885, −3.42208860322518367128159568356, −3.14099252249708479365203334680, −3.02028627755577217815547505454, −2.74949312975099090724305925870, −2.58701381403398510692258853420, −2.57490836710793846027632685661, −2.54119648073484647939638406202, −1.77809723784049884372201133048, −1.58519145803084804175052005472, −1.28709742138767331941989284185, −1.19369179840054501224239449699, −1.07603075413844062980935881342, −0.867995735331923963404615627086, −0.38101654141231479877230176497, 0.38101654141231479877230176497, 0.867995735331923963404615627086, 1.07603075413844062980935881342, 1.19369179840054501224239449699, 1.28709742138767331941989284185, 1.58519145803084804175052005472, 1.77809723784049884372201133048, 2.54119648073484647939638406202, 2.57490836710793846027632685661, 2.58701381403398510692258853420, 2.74949312975099090724305925870, 3.02028627755577217815547505454, 3.14099252249708479365203334680, 3.42208860322518367128159568356, 3.72145895952480140282974803885, 3.80587243408054002686252434227, 3.89020791889088343854179947962, 4.23275347176796323784113961632, 4.42408101901510414349463888895, 4.53735514978530019890637063504, 4.65289890291381468538172867432, 4.74233028439509593182627061415, 4.89397526299315625053911525482, 5.01478102127000862832057924351, 5.35813398189847867956567741615

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

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