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Properties

Label 12-2738e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.213\times 10^{20}$
Sign $1$
Analytic cond. $1.09210\times 10^{8}$
Root an. cond. $4.67579$
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  + 6·2-s + 21·4-s − 3·7-s + 56·8-s − 3·9-s − 3·11-s − 18·14-s + 126·16-s + 3·17-s − 18·18-s + 21·19-s − 18·22-s + 21·23-s − 24·25-s − 27-s − 63·28-s − 6·29-s + 21·31-s + 252·32-s + 18·34-s − 63·36-s + 126·38-s + 18·41-s + 18·43-s − 63·44-s + 126·46-s − 9·47-s + ⋯
L(s)  = 1  + 4.24·2-s + 21/2·4-s − 1.13·7-s + 19.7·8-s − 9-s − 0.904·11-s − 4.81·14-s + 63/2·16-s + 0.727·17-s − 4.24·18-s + 4.81·19-s − 3.83·22-s + 4.37·23-s − 4.79·25-s − 0.192·27-s − 11.9·28-s − 1.11·29-s + 3.77·31-s + 44.5·32-s + 3.08·34-s − 10.5·36-s + 20.4·38-s + 2.81·41-s + 2.74·43-s − 9.49·44-s + 18.5·46-s − 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(1.09210\times 10^{8}\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 37^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(360.0984989\)
\(L(\frac12)\) \(\approx\) \(360.0984989\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
37 \( 1 \)
good3 \( 1 + p T^{2} + T^{3} + 5 p T^{4} + p^{2} T^{5} + 26 T^{6} + p^{3} T^{7} + 5 p^{3} T^{8} + p^{3} T^{9} + p^{5} T^{10} + p^{6} T^{12} \)
5 \( ( 1 + 12 T^{2} + T^{3} + 12 p T^{4} + p^{3} T^{6} )^{2} \)
7 \( 1 + 3 T + 18 T^{2} + 68 T^{3} + 33 p T^{4} + 645 T^{5} + 2092 T^{6} + 645 p T^{7} + 33 p^{3} T^{8} + 68 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T + 42 T^{2} + 128 T^{3} + 81 p T^{4} + 2505 T^{5} + 12036 T^{6} + 2505 p T^{7} + 81 p^{3} T^{8} + 128 p^{3} T^{9} + 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} - 43 T^{3} + 675 T^{4} - 1494 T^{5} + 9057 T^{6} - 1494 p T^{7} + 675 p^{2} T^{8} - 43 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 3 T + 78 T^{2} - 186 T^{3} + 2823 T^{4} - 5541 T^{5} + 60887 T^{6} - 5541 p T^{7} + 2823 p^{2} T^{8} - 186 p^{3} T^{9} + 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 21 T + 270 T^{2} - 2448 T^{3} + 921 p T^{4} - 100959 T^{5} + 483804 T^{6} - 100959 p T^{7} + 921 p^{3} T^{8} - 2448 p^{3} T^{9} + 270 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 21 T + 282 T^{2} - 2750 T^{3} + 21195 T^{4} - 133917 T^{5} + 701012 T^{6} - 133917 p T^{7} + 21195 p^{2} T^{8} - 2750 p^{3} T^{9} + 282 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 150 T^{2} + 786 T^{3} + 10140 T^{4} + 42924 T^{5} + 384635 T^{6} + 42924 p T^{7} + 10140 p^{2} T^{8} + 786 p^{3} T^{9} + 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 21 T + 210 T^{2} - 1538 T^{3} + 11139 T^{4} - 78321 T^{5} + 476692 T^{6} - 78321 p T^{7} + 11139 p^{2} T^{8} - 1538 p^{3} T^{9} + 210 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 18 T + 219 T^{2} - 1715 T^{3} + 10905 T^{4} - 53295 T^{5} + 317382 T^{6} - 53295 p T^{7} + 10905 p^{2} T^{8} - 1715 p^{3} T^{9} + 219 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 309 T^{2} - 3231 T^{3} + 32763 T^{4} - 247509 T^{5} + 1842654 T^{6} - 247509 p T^{7} + 32763 p^{2} T^{8} - 3231 p^{3} T^{9} + 309 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T + 3 p T^{2} + 1090 T^{3} + 12879 T^{4} + 83373 T^{5} + 712326 T^{6} + 83373 p T^{7} + 12879 p^{2} T^{8} + 1090 p^{3} T^{9} + 3 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 18 T + 438 T^{2} - 5131 T^{3} + 68049 T^{4} - 563244 T^{5} + 5050281 T^{6} - 563244 p T^{7} + 68049 p^{2} T^{8} - 5131 p^{3} T^{9} + 438 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 27 T + 630 T^{2} - 9304 T^{3} + 120531 T^{4} - 1180233 T^{5} + 10248444 T^{6} - 1180233 p T^{7} + 120531 p^{2} T^{8} - 9304 p^{3} T^{9} + 630 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 24 T + 435 T^{2} + 5621 T^{3} + 65175 T^{4} + 611787 T^{5} + 5220970 T^{6} + 611787 p T^{7} + 65175 p^{2} T^{8} + 5621 p^{3} T^{9} + 435 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T + 258 T^{2} - 2088 T^{3} + 31863 T^{4} - 215955 T^{5} + 2554332 T^{6} - 215955 p T^{7} + 31863 p^{2} T^{8} - 2088 p^{3} T^{9} + 258 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 318 T^{2} + 72 T^{3} + 46479 T^{4} + 16488 T^{5} + 4109252 T^{6} + 16488 p T^{7} + 46479 p^{2} T^{8} + 72 p^{3} T^{9} + 318 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 27 T + 546 T^{2} - 7362 T^{3} + 89157 T^{4} - 878589 T^{5} + 8191977 T^{6} - 878589 p T^{7} + 89157 p^{2} T^{8} - 7362 p^{3} T^{9} + 546 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 21 T + 537 T^{2} - 7554 T^{3} + 110715 T^{4} - 1137153 T^{5} + 11786006 T^{6} - 1137153 p T^{7} + 110715 p^{2} T^{8} - 7554 p^{3} T^{9} + 537 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 27 T + 774 T^{2} - 12544 T^{3} + 198111 T^{4} - 2196657 T^{5} + 23366916 T^{6} - 2196657 p T^{7} + 198111 p^{2} T^{8} - 12544 p^{3} T^{9} + 774 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 21 T + 459 T^{2} + 6576 T^{3} + 94140 T^{4} + 1013100 T^{5} + 10675961 T^{6} + 1013100 p T^{7} + 94140 p^{2} T^{8} + 6576 p^{3} T^{9} + 459 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 42 T + 1131 T^{2} - 22535 T^{3} + 354777 T^{4} - 4575915 T^{5} + 49371670 T^{6} - 4575915 p T^{7} + 354777 p^{2} T^{8} - 22535 p^{3} T^{9} + 1131 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.60272744519567110015619708250, −4.47131013662393370992690885536, −4.17702763883323041864582476227, −4.13334787497297578444515970465, −4.11762592653758361432272983653, −3.65936046101162921097059661451, −3.65910843082896368476659406599, −3.41318501251781655326701362352, −3.33947242051495399610783675521, −3.31002078604316722507054260797, −3.23165734573108811563818624829, −3.21134260863947954665160422288, −2.86419344133991169010427846215, −2.65875190408075277290824652207, −2.49868633175519812193845113511, −2.33587664782211091778587878752, −2.26361945636921679297954379357, −2.20563395341400762499359665190, −2.07074291301131409229417896247, −1.48284063988042639018846749198, −1.27773793632809221690416142247, −1.05888475461042723287318390004, −0.836519248136080886170989545859, −0.73297894398106217141304618874, −0.64700941864325567050253832360, 0.64700941864325567050253832360, 0.73297894398106217141304618874, 0.836519248136080886170989545859, 1.05888475461042723287318390004, 1.27773793632809221690416142247, 1.48284063988042639018846749198, 2.07074291301131409229417896247, 2.20563395341400762499359665190, 2.26361945636921679297954379357, 2.33587664782211091778587878752, 2.49868633175519812193845113511, 2.65875190408075277290824652207, 2.86419344133991169010427846215, 3.21134260863947954665160422288, 3.23165734573108811563818624829, 3.31002078604316722507054260797, 3.33947242051495399610783675521, 3.41318501251781655326701362352, 3.65910843082896368476659406599, 3.65936046101162921097059661451, 4.11762592653758361432272983653, 4.13334787497297578444515970465, 4.17702763883323041864582476227, 4.47131013662393370992690885536, 4.60272744519567110015619708250

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

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