Properties

Label 12-912e6-1.1-c2e6-0-0
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $2.35493\times 10^{8}$
Root an. cond. $4.98499$
Motivic weight $2$
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  − 2·5-s + 2·7-s − 9·9-s + 26·11-s − 50·17-s + 10·19-s − 28·23-s − 59·25-s − 4·35-s + 210·43-s + 18·45-s − 22·47-s − 163·49-s − 52·55-s + 214·61-s − 18·63-s + 102·73-s + 52·77-s + 54·81-s + 404·83-s + 100·85-s − 20·95-s − 234·99-s − 164·101-s + 56·115-s − 100·119-s − 203·121-s + ⋯
L(s)  = 1  − 2/5·5-s + 2/7·7-s − 9-s + 2.36·11-s − 2.94·17-s + 0.526·19-s − 1.21·23-s − 2.35·25-s − 0.114·35-s + 4.88·43-s + 2/5·45-s − 0.468·47-s − 3.32·49-s − 0.945·55-s + 3.50·61-s − 2/7·63-s + 1.39·73-s + 0.675·77-s + 2/3·81-s + 4.86·83-s + 1.17·85-s − 0.210·95-s − 2.36·99-s − 1.62·101-s + 0.486·115-s − 0.840·119-s − 1.67·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2.35493\times 10^{8}\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.273135765\)
\(L(\frac12)\) \(\approx\) \(2.273135765\)
\(L(2)\) 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 \( ( 1 + p T^{2} )^{3} \)
19 \( 1 - 10 T + 407 T^{2} - 388 p T^{3} + 407 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} \)
good5 \( ( 1 + T + 31 T^{2} + 74 T^{3} + 31 p^{2} T^{4} + p^{4} T^{5} + p^{6} T^{6} )^{2} \)
7 \( ( 1 - T + 83 T^{2} - 262 T^{3} + 83 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( ( 1 - 13 T + 355 T^{2} - 3134 T^{3} + 355 p^{2} T^{4} - 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
13 \( 1 - 222 T^{2} - 8913 T^{4} + 8535292 T^{6} - 8913 p^{4} T^{8} - 222 p^{8} T^{10} + p^{12} T^{12} \)
17 \( ( 1 + 25 T + 835 T^{2} + 14342 T^{3} + 835 p^{2} T^{4} + 25 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
23 \( ( 1 + 14 T + 1027 T^{2} + 9676 T^{3} + 1027 p^{2} T^{4} + 14 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
29 \( 1 - 1878 T^{2} + 1963455 T^{4} - 1513144244 T^{6} + 1963455 p^{4} T^{8} - 1878 p^{8} T^{10} + p^{12} T^{12} \)
31 \( 1 - 2574 T^{2} + 2592063 T^{4} - 2002604132 T^{6} + 2592063 p^{4} T^{8} - 2574 p^{8} T^{10} + p^{12} T^{12} \)
37 \( 1 - 2238 T^{2} + 4271055 T^{4} - 7604589764 T^{6} + 4271055 p^{4} T^{8} - 2238 p^{8} T^{10} + p^{12} T^{12} \)
41 \( 1 + 810 T^{2} + 4444335 T^{4} + 84590476 T^{6} + 4444335 p^{4} T^{8} + 810 p^{8} T^{10} + p^{12} T^{12} \)
43 \( ( 1 - 105 T + 8979 T^{2} - 424054 T^{3} + 8979 p^{2} T^{4} - 105 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
47 \( ( 1 + 11 T + 4543 T^{2} + 23326 T^{3} + 4543 p^{2} T^{4} + 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
53 \( 1 - 12582 T^{2} + 74706975 T^{4} - 265412742932 T^{6} + 74706975 p^{4} T^{8} - 12582 p^{8} T^{10} + p^{12} T^{12} \)
59 \( 1 - 1350 T^{2} + 25160031 T^{4} - 18935070932 T^{6} + 25160031 p^{4} T^{8} - 1350 p^{8} T^{10} + p^{12} T^{12} \)
61 \( ( 1 - 107 T + 11099 T^{2} - 692018 T^{3} + 11099 p^{2} T^{4} - 107 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
67 \( 1 - 22614 T^{2} + 229583679 T^{4} - 1330212756404 T^{6} + 229583679 p^{4} T^{8} - 22614 p^{8} T^{10} + p^{12} T^{12} \)
71 \( 1 - 18774 T^{2} + 191507823 T^{4} - 1184211826292 T^{6} + 191507823 p^{4} T^{8} - 18774 p^{8} T^{10} + p^{12} T^{12} \)
73 \( ( 1 - 51 T + 8091 T^{2} - 428066 T^{3} + 8091 p^{2} T^{4} - 51 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
79 \( 1 - 17598 T^{2} + 157161279 T^{4} - 1016250978692 T^{6} + 157161279 p^{4} T^{8} - 17598 p^{8} T^{10} + p^{12} T^{12} \)
83 \( ( 1 - 202 T + 33559 T^{2} - 3043148 T^{3} + 33559 p^{2} T^{4} - 202 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
89 \( 1 - 44118 T^{2} + 836433135 T^{4} - 8707655419508 T^{6} + 836433135 p^{4} T^{8} - 44118 p^{8} T^{10} + p^{12} T^{12} \)
97 \( 1 - 48054 T^{2} + 1028309583 T^{4} - 12508133393972 T^{6} + 1028309583 p^{4} T^{8} - 48054 p^{8} T^{10} + p^{12} T^{12} \)
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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

−5.11016177048031188976823270480, −4.99657632661111438120043478562, −4.62088083565055306769562429669, −4.59728290255258650419009357888, −4.58597362611814094060812887116, −4.13733267072763275381167681072, −4.09737785632460629895641032874, −3.94866666676744572030419100220, −3.74237359072322690330237911832, −3.67612443761331536669619531352, −3.56557259204351353804002339593, −3.55283962769802572107502143808, −2.82330410639790283968937586084, −2.74049306947625592249865551901, −2.73097793556909947738090495677, −2.31760745744604812261480502256, −2.28193511016695610222657776718, −1.93591309105711078343027722240, −1.93494832317596167040406530906, −1.67889518065885754144592047254, −1.27430406206166441442043000818, −0.951178901514835488205866721933, −0.856299091513999629121019094538, −0.34778027883780155609669672267, −0.23333296745769800979710203547, 0.23333296745769800979710203547, 0.34778027883780155609669672267, 0.856299091513999629121019094538, 0.951178901514835488205866721933, 1.27430406206166441442043000818, 1.67889518065885754144592047254, 1.93494832317596167040406530906, 1.93591309105711078343027722240, 2.28193511016695610222657776718, 2.31760745744604812261480502256, 2.73097793556909947738090495677, 2.74049306947625592249865551901, 2.82330410639790283968937586084, 3.55283962769802572107502143808, 3.56557259204351353804002339593, 3.67612443761331536669619531352, 3.74237359072322690330237911832, 3.94866666676744572030419100220, 4.09737785632460629895641032874, 4.13733267072763275381167681072, 4.58597362611814094060812887116, 4.59728290255258650419009357888, 4.62088083565055306769562429669, 4.99657632661111438120043478562, 5.11016177048031188976823270480

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

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