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Properties

Label 12-21e12-1.1-c1e6-0-4
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
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 + 6·4-s + 3·5-s − 9·8-s − 9·10-s − 6·11-s − 3·13-s + 12·16-s − 12·17-s + 6·19-s + 18·20-s + 18·22-s − 12·23-s + 15·25-s + 9·26-s − 9·27-s − 9·29-s − 3·31-s − 12·32-s + 36·34-s − 6·37-s − 18·38-s − 27·40-s + 3·43-s − 36·44-s + 36·46-s + 3·47-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s + 1.34·5-s − 3.18·8-s − 2.84·10-s − 1.80·11-s − 0.832·13-s + 3·16-s − 2.91·17-s + 1.37·19-s + 4.02·20-s + 3.83·22-s − 2.50·23-s + 3·25-s + 1.76·26-s − 1.73·27-s − 1.67·29-s − 0.538·31-s − 2.12·32-s + 6.17·34-s − 0.986·37-s − 2.91·38-s − 4.26·40-s + 0.457·43-s − 5.42·44-s + 5.30·46-s + 0.437·47-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6859937569\)
\(L(\frac12)\) \(\approx\) \(0.6859937569\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{2} T^{3} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - 11 T^{6} - 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 852 p T^{7} + 492 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 3 T^{2} + 76 T^{3} + 45 T^{4} - 135 T^{5} + 3246 T^{6} - 135 p T^{7} + 45 p^{2} T^{8} + 76 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 6060 p T^{7} + 420 p^{2} T^{8} + 54 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 1962 p T^{7} + 171 p^{2} T^{8} + 319 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 11064 p T^{7} + 2481 p^{2} T^{8} + 405 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 15798 p T^{7} + 3615 p^{2} T^{8} - 495 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 132 T^{2} + 418 T^{3} + 13698 T^{4} - 19134 T^{5} - 893289 T^{6} - 19134 p T^{7} + 13698 p^{2} T^{8} + 418 p^{3} T^{9} - 132 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 50094 p T^{7} + 15318 p^{2} T^{8} + 518 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 21 T + 303 T^{2} - 2797 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 195678 p T^{7} + 25767 p^{2} T^{8} - 499 p^{3} T^{9} + 84 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 2421 T^{4} - 11502 T^{5} + 340233 T^{6} - 11502 p T^{7} + 2421 p^{2} T^{8} - 149 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.08326337723307081809231840340, −5.67844288519085951184998106691, −5.66869753929676337997398512140, −5.62314904051955695844504743676, −5.29910962451854833512734116364, −5.20956273626953559492910596511, −5.13935601407179292856399639234, −5.04352777388769430097390418726, −4.49201352824221679105927262574, −4.27156308938759156736074494962, −4.26779620305591964594936952992, −3.85946546189282595522289114244, −3.73886746560442054309360081203, −3.54394925689353323755237639096, −3.09027669157189502421267644268, −3.03559974975142234966392242999, −2.48036989865594839413021645218, −2.39409908794547250465066989338, −2.38824226298491302348913423670, −2.04411221979611900259145158075, −1.96793657960726988392560763943, −1.77658191995405483126933292561, −1.25276497863972632445077942768, −0.58166333748627541630779119806, −0.43487204845687675740811970151, 0.43487204845687675740811970151, 0.58166333748627541630779119806, 1.25276497863972632445077942768, 1.77658191995405483126933292561, 1.96793657960726988392560763943, 2.04411221979611900259145158075, 2.38824226298491302348913423670, 2.39409908794547250465066989338, 2.48036989865594839413021645218, 3.03559974975142234966392242999, 3.09027669157189502421267644268, 3.54394925689353323755237639096, 3.73886746560442054309360081203, 3.85946546189282595522289114244, 4.26779620305591964594936952992, 4.27156308938759156736074494962, 4.49201352824221679105927262574, 5.04352777388769430097390418726, 5.13935601407179292856399639234, 5.20956273626953559492910596511, 5.29910962451854833512734116364, 5.62314904051955695844504743676, 5.66869753929676337997398512140, 5.67844288519085951184998106691, 6.08326337723307081809231840340

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

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