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Properties

Label 12-4016e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.195\times 10^{21}$
Sign $1$
Analytic cond. $1.08748\times 10^{9}$
Root an. cond. $5.66285$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 6·7-s − 9-s − 11-s − 5·13-s + 15-s + 8·17-s − 3·19-s + 6·21-s − 18·23-s − 15·25-s − 3·27-s + 29-s − 6·31-s + 33-s + 6·35-s − 13·37-s + 5·39-s + 4·41-s + 5·43-s + 45-s − 8·47-s + 49-s − 8·51-s − 3·53-s + 55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2.26·7-s − 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s + 1.94·17-s − 0.688·19-s + 1.30·21-s − 3.75·23-s − 3·25-s − 0.577·27-s + 0.185·29-s − 1.07·31-s + 0.174·33-s + 1.01·35-s − 2.13·37-s + 0.800·39-s + 0.624·41-s + 0.762·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 1.12·51-s − 0.412·53-s + 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 251^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 251^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 251^{6}\)
Sign: $1$
Analytic conductor: \(1.08748\times 10^{9}\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 251^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( ( 1 - T )^{6} \)
good3 \( 1 + T + 2 T^{2} + 2 p T^{3} + 17 T^{4} + 17 T^{5} + 32 T^{6} + 17 p T^{7} + 17 p^{2} T^{8} + 2 p^{4} T^{9} + 2 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + T + 16 T^{2} + 22 T^{3} + 139 T^{4} + 169 T^{5} + 848 T^{6} + 169 p T^{7} + 139 p^{2} T^{8} + 22 p^{3} T^{9} + 16 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 5 p T^{2} + 120 T^{3} + 393 T^{4} + 978 T^{5} + 2747 T^{6} + 978 p T^{7} + 393 p^{2} T^{8} + 120 p^{3} T^{9} + 5 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T + 36 T^{2} + 32 T^{3} + 775 T^{4} + 563 T^{5} + 10168 T^{6} + 563 p T^{7} + 775 p^{2} T^{8} + 32 p^{3} T^{9} + 36 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 5 T + 54 T^{2} + 184 T^{3} + 93 p T^{4} + 3311 T^{5} + 17968 T^{6} + 3311 p T^{7} + 93 p^{3} T^{8} + 184 p^{3} T^{9} + 54 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + 93 T^{2} - 556 T^{3} + 3569 T^{4} - 16744 T^{5} + 77417 T^{6} - 16744 p T^{7} + 3569 p^{2} T^{8} - 556 p^{3} T^{9} + 93 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T + 68 T^{2} + 156 T^{3} + 1985 T^{4} + 3861 T^{5} + 40284 T^{6} + 3861 p T^{7} + 1985 p^{2} T^{8} + 156 p^{3} T^{9} + 68 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 18 T + 233 T^{2} + 2096 T^{3} + 15827 T^{4} + 4176 p T^{5} + 506401 T^{6} + 4176 p^{2} T^{7} + 15827 p^{2} T^{8} + 2096 p^{3} T^{9} + 233 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - T + 28 T^{2} + 16 T^{3} + 687 T^{4} + 1345 T^{5} + 5872 T^{6} + 1345 p T^{7} + 687 p^{2} T^{8} + 16 p^{3} T^{9} + 28 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 6 T + 119 T^{2} + 684 T^{3} + 7365 T^{4} + 37394 T^{5} + 281235 T^{6} + 37394 p T^{7} + 7365 p^{2} T^{8} + 684 p^{3} T^{9} + 119 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 13 T + 250 T^{2} + 2314 T^{3} + 24501 T^{4} + 167949 T^{5} + 1229888 T^{6} + 167949 p T^{7} + 24501 p^{2} T^{8} + 2314 p^{3} T^{9} + 250 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 149 T^{2} - 476 T^{3} + 9989 T^{4} - 26584 T^{5} + 453769 T^{6} - 26584 p T^{7} + 9989 p^{2} T^{8} - 476 p^{3} T^{9} + 149 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 5 T + 106 T^{2} - 152 T^{3} + 5533 T^{4} + 4289 T^{5} + 220872 T^{6} + 4289 p T^{7} + 5533 p^{2} T^{8} - 152 p^{3} T^{9} + 106 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 8 T + 194 T^{2} + 1024 T^{3} + 16399 T^{4} + 66232 T^{5} + 908508 T^{6} + 66232 p T^{7} + 16399 p^{2} T^{8} + 1024 p^{3} T^{9} + 194 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 3 T + 4 p T^{2} + 824 T^{3} + 21599 T^{4} + 87165 T^{5} + 1392784 T^{6} + 87165 p T^{7} + 21599 p^{2} T^{8} + 824 p^{3} T^{9} + 4 p^{5} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 5 T + 122 T^{2} - 96 T^{3} + 7917 T^{4} + 4213 T^{5} + 555224 T^{6} + 4213 p T^{7} + 7917 p^{2} T^{8} - 96 p^{3} T^{9} + 122 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + p T + 1886 T^{2} + 38382 T^{3} + 568285 T^{4} + 6417813 T^{5} + 56554408 T^{6} + 6417813 p T^{7} + 568285 p^{2} T^{8} + 38382 p^{3} T^{9} + 1886 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
67 \( 1 - 13 T + 156 T^{2} - 1026 T^{3} + 10455 T^{4} - 87493 T^{5} + 1010848 T^{6} - 87493 p T^{7} + 10455 p^{2} T^{8} - 1026 p^{3} T^{9} + 156 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 22 T + 398 T^{2} + 5466 T^{3} + 64191 T^{4} + 655668 T^{5} + 5763460 T^{6} + 655668 p T^{7} + 64191 p^{2} T^{8} + 5466 p^{3} T^{9} + 398 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 6 T + 155 T^{2} - 1180 T^{3} + 23493 T^{4} - 140690 T^{5} + 1759919 T^{6} - 140690 p T^{7} + 23493 p^{2} T^{8} - 1180 p^{3} T^{9} + 155 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 627 T^{2} + 9560 T^{3} + 125279 T^{4} + 1350546 T^{5} + 12950923 T^{6} + 1350546 p T^{7} + 125279 p^{2} T^{8} + 9560 p^{3} T^{9} + 627 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 14 T + 246 T^{2} + 1978 T^{3} + 28727 T^{4} + 273460 T^{5} + 3322100 T^{6} + 273460 p T^{7} + 28727 p^{2} T^{8} + 1978 p^{3} T^{9} + 246 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 339 T^{2} - 2768 T^{3} + 23317 T^{4} - 2718 T^{5} + 223443 T^{6} - 2718 p T^{7} + 23317 p^{2} T^{8} - 2768 p^{3} T^{9} + 339 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 258 T^{2} - 1728 T^{3} + 15615 T^{4} + 49936 T^{5} - 330564 T^{6} + 49936 p T^{7} + 15615 p^{2} T^{8} - 1728 p^{3} T^{9} + 258 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.75937255632728757726065199049, −4.66625554961966364102950547233, −4.47634182155434909577028678410, −4.12982321476505406263580544927, −4.07086840610606255789308052961, −4.06197691073275482021932654504, −3.97654176814884240616066641238, −3.84822977270839381840951715935, −3.50884257582252310974205195605, −3.50070609781604134514587559619, −3.35149896101831109245089683651, −3.34789904385968693901449560346, −3.09990104190750431966396989302, −2.79617869580730157934546795001, −2.63917676638376755326165028458, −2.58435521603116061701347941452, −2.53826661927339437064877996362, −2.48306317950510398312398647289, −1.81591135089856087636516429464, −1.75435273192840656288073561832, −1.73950682364969971130318672364, −1.63243698416375054402107691020, −1.51994063905283880210585474241, −1.21776097376954244644323103062, −0.848840689806556500712799689157, 0, 0, 0, 0, 0, 0, 0.848840689806556500712799689157, 1.21776097376954244644323103062, 1.51994063905283880210585474241, 1.63243698416375054402107691020, 1.73950682364969971130318672364, 1.75435273192840656288073561832, 1.81591135089856087636516429464, 2.48306317950510398312398647289, 2.53826661927339437064877996362, 2.58435521603116061701347941452, 2.63917676638376755326165028458, 2.79617869580730157934546795001, 3.09990104190750431966396989302, 3.34789904385968693901449560346, 3.35149896101831109245089683651, 3.50070609781604134514587559619, 3.50884257582252310974205195605, 3.84822977270839381840951715935, 3.97654176814884240616066641238, 4.06197691073275482021932654504, 4.07086840610606255789308052961, 4.12982321476505406263580544927, 4.47634182155434909577028678410, 4.66625554961966364102950547233, 4.75937255632728757726065199049

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

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