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Properties

Label 12-21e12-1.1-c1e6-0-5
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 2·4-s − 10·5-s + 2·6-s − 8-s + 6·9-s − 10·10-s − 4·11-s + 4·12-s − 3·13-s − 20·15-s + 12·17-s + 6·18-s + 3·19-s − 20·20-s − 4·22-s − 2·24-s + 41·25-s − 3·26-s + 7·27-s − 29-s − 20·30-s + 3·31-s − 4·32-s − 8·33-s + 12·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 4-s − 4.47·5-s + 0.816·6-s − 0.353·8-s + 2·9-s − 3.16·10-s − 1.20·11-s + 1.15·12-s − 0.832·13-s − 5.16·15-s + 2.91·17-s + 1.41·18-s + 0.688·19-s − 4.47·20-s − 0.852·22-s − 0.408·24-s + 41/5·25-s − 0.588·26-s + 1.34·27-s − 0.185·29-s − 3.65·30-s + 0.538·31-s − 0.707·32-s − 1.39·33-s + 2.05·34-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

\(L(1)\) \(\approx\) \(2.718310108\)
\(L(\frac12)\) \(\approx\) \(2.718310108\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 2 T - 2 T^{2} + p^{2} T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 - T - T^{2} + p^{2} T^{3} - 3 T^{4} - p T^{5} + 13 T^{6} - p^{2} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + p T + 17 T^{2} + 39 T^{3} + 17 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 2 T + 14 T^{2} - 3 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 12 T + 54 T^{2} - 210 T^{3} + 1350 T^{4} - 5898 T^{5} + 19735 T^{6} - 5898 p T^{7} + 1350 p^{2} T^{8} - 210 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 61 T^{3} + 69 p T^{4} - 726 T^{5} - 27501 T^{6} - 726 p T^{7} + 69 p^{3} T^{8} + 61 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 36 T^{2} - 9 T^{3} + 36 p T^{4} + p^{3} T^{6} )^{2} \)
29 \( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 758 p T^{7} + 4425 p^{2} T^{8} - 31 p^{3} T^{9} - 82 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 4746 p T^{7} + 1983 p^{2} T^{8} + 219 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 135116 p T^{7} + 18432 p^{2} T^{8} - 1802 p^{3} T^{9} + 206 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 18 T + 90 T^{2} - 378 T^{3} + 7848 T^{4} - 52668 T^{5} + 160459 T^{6} - 52668 p T^{7} + 7848 p^{2} T^{8} - 378 p^{3} T^{9} + 90 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 20556 p T^{7} + 10161 p^{2} T^{8} + 459 p^{3} T^{9} - 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 11472 p T^{7} + 12372 p^{2} T^{8} + 358 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 3 T - 42 T^{2} - 1209 T^{3} - 3165 T^{4} + 28380 T^{5} + 1003961 T^{6} + 28380 p T^{7} - 3165 p^{2} T^{8} - 1209 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 90444 p T^{7} + 34812 p^{2} T^{8} + 582 p^{3} T^{9} - 144 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 2 T - 112 T^{2} + 1238 T^{3} + 1662 T^{4} - 59806 T^{5} + 720895 T^{6} - 59806 p T^{7} + 1662 p^{2} T^{8} + 1238 p^{3} T^{9} - 112 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.90008504526266013016640516334, −5.62887096704238844682589655667, −5.57770760351181375658980786695, −5.56962810906274523851419656299, −5.33608891328725984526118772991, −5.16065866695518696909361213231, −4.75269203355369190188989303956, −4.70108058227496479164179504753, −4.10073594017250321008704666569, −4.09406272915861283726589052920, −4.05611834620167493487253016992, −4.03554219161213524718551211512, −4.01010783249070583872184249446, −3.70658191472231199285768690279, −3.27410587270251890043924365850, −3.17285578977244598190542924527, −2.97232722948196740474873912056, −2.90252439774174758763314600069, −2.62301239629133119775779063264, −2.33023948727570362732917775114, −2.15880911901928764762469436026, −1.62998602028642733219654517237, −1.00126377297477714791460656952, −0.887013104025779839170708036383, −0.44495275735223829949425951744, 0.44495275735223829949425951744, 0.887013104025779839170708036383, 1.00126377297477714791460656952, 1.62998602028642733219654517237, 2.15880911901928764762469436026, 2.33023948727570362732917775114, 2.62301239629133119775779063264, 2.90252439774174758763314600069, 2.97232722948196740474873912056, 3.17285578977244598190542924527, 3.27410587270251890043924365850, 3.70658191472231199285768690279, 4.01010783249070583872184249446, 4.03554219161213524718551211512, 4.05611834620167493487253016992, 4.09406272915861283726589052920, 4.10073594017250321008704666569, 4.70108058227496479164179504753, 4.75269203355369190188989303956, 5.16065866695518696909361213231, 5.33608891328725984526118772991, 5.56962810906274523851419656299, 5.57770760351181375658980786695, 5.62887096704238844682589655667, 5.90008504526266013016640516334

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

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