Properties

Label 12-912e6-1.1-c1e6-0-6
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  − 3·3-s − 6·7-s + 3·9-s + 12·11-s − 3·13-s + 18·21-s + 3·25-s + 2·27-s − 12·29-s − 30·31-s − 36·33-s − 6·37-s + 9·39-s − 12·41-s + 9·43-s − 18·47-s + 3·49-s − 6·59-s + 3·61-s − 18·63-s − 3·67-s + 6·71-s + 9·73-s − 9·75-s − 72·77-s + 27·79-s − 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s − 2.26·7-s + 9-s + 3.61·11-s − 0.832·13-s + 3.92·21-s + 3/5·25-s + 0.384·27-s − 2.22·29-s − 5.38·31-s − 6.26·33-s − 0.986·37-s + 1.44·39-s − 1.87·41-s + 1.37·43-s − 2.62·47-s + 3/7·49-s − 0.781·59-s + 0.384·61-s − 2.26·63-s − 0.366·67-s + 0.712·71-s + 1.05·73-s − 1.03·75-s − 8.20·77-s + 3.03·79-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5027874142\)
\(L(\frac12)\) \(\approx\) \(0.5027874142\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T + T^{2} )^{3} \)
19 \( 1 + 9 T^{2} - 64 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
good5 \( 1 - 3 T^{2} + 16 T^{3} - 6 T^{4} - 24 T^{5} + 269 T^{6} - 24 p T^{7} - 6 p^{2} T^{8} + 16 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
7 \( ( 1 + 3 T + 12 T^{2} + 39 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 6 T + 9 T^{2} + 4 T^{3} + 9 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 3 T^{2} + 128 T^{3} - 42 T^{4} - 192 T^{5} + 13769 T^{6} - 192 p T^{7} - 42 p^{2} T^{8} + 128 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 57 T^{2} - 16 T^{3} + 1938 T^{4} + 456 T^{5} - 50329 T^{6} + 456 p T^{7} + 1938 p^{2} T^{8} - 16 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 12 T + 57 T^{2} + 36 T^{3} - 1002 T^{4} - 228 p T^{5} - 37811 T^{6} - 228 p^{2} T^{7} - 1002 p^{2} T^{8} + 36 p^{3} T^{9} + 57 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 15 T + 132 T^{2} + 803 T^{3} + 132 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 3 T + 66 T^{2} + 3 p T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 12 T + 21 T^{2} - 108 T^{3} + 582 T^{4} - 84 p T^{5} - 97247 T^{6} - 84 p^{2} T^{7} + 582 p^{2} T^{8} - 108 p^{3} T^{9} + 21 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T - 63 T^{2} + 218 T^{3} + 7731 T^{4} - 14193 T^{5} - 309354 T^{6} - 14193 p T^{7} + 7731 p^{2} T^{8} + 218 p^{3} T^{9} - 63 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \)
53 \( 1 - 75 T^{2} + 592 T^{3} + 1650 T^{4} - 22200 T^{5} + 87245 T^{6} - 22200 p T^{7} + 1650 p^{2} T^{8} + 592 p^{3} T^{9} - 75 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 6 T - 45 T^{2} - 82 T^{3} + 78 T^{4} - 13458 T^{5} - 21001 T^{6} - 13458 p T^{7} + 78 p^{2} T^{8} - 82 p^{3} T^{9} - 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 129 T^{2} + 352 T^{3} + 9477 T^{4} - 13509 T^{5} - 594522 T^{6} - 13509 p T^{7} + 9477 p^{2} T^{8} + 352 p^{3} T^{9} - 129 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 159 T^{2} - 374 T^{3} + 15651 T^{4} + 19683 T^{5} - 1137162 T^{6} + 19683 p T^{7} + 15651 p^{2} T^{8} - 374 p^{3} T^{9} - 159 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T - 45 T^{2} - 494 T^{3} + 834 T^{4} + 39090 T^{5} + 68531 T^{6} + 39090 p T^{7} + 834 p^{2} T^{8} - 494 p^{3} T^{9} - 45 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 9 T - 117 T^{2} + 484 T^{3} + 14553 T^{4} - 12123 T^{5} - 1231818 T^{6} - 12123 p T^{7} + 14553 p^{2} T^{8} + 484 p^{3} T^{9} - 117 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 27 T + 333 T^{2} - 2806 T^{3} + 19071 T^{4} - 40599 T^{5} - 404826 T^{6} - 40599 p T^{7} + 19071 p^{2} T^{8} - 2806 p^{3} T^{9} + 333 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 12 T + 153 T^{2} - 2056 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 18 T - 15 T^{2} - 162 T^{3} + 28374 T^{4} + 118170 T^{5} - 1575011 T^{6} + 118170 p T^{7} + 28374 p^{2} T^{8} - 162 p^{3} T^{9} - 15 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T - 123 T^{2} + 338 T^{3} + 8010 T^{4} - 79650 T^{5} - 870771 T^{6} - 79650 p T^{7} + 8010 p^{2} T^{8} + 338 p^{3} T^{9} - 123 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.31165621003786517263243180217, −5.30188113725349336541977541325, −5.20283934819457970387306585801, −4.97817212847626108765272445169, −4.74141306071808377687636105008, −4.58938102270009655713289899411, −4.35946617545032578884051583535, −4.06613190895187954541176222906, −3.84967556953359069040535674931, −3.75376033726628338931156099432, −3.68291326029135179920936072981, −3.62120021017033561248461954104, −3.36009794106497061505609984199, −3.31605606642929933631129634443, −3.01426929548178601615416481343, −2.79326782988793334970850667669, −2.51128029913974098750449636693, −1.99448560552029089234674398879, −1.85330441331705195938803624285, −1.79808535618597171811345386664, −1.53726452050467517688175607532, −1.49698808146477888443887941193, −0.794530164684294562167648622508, −0.36135097070466020634331333488, −0.31115395571488640387238371104, 0.31115395571488640387238371104, 0.36135097070466020634331333488, 0.794530164684294562167648622508, 1.49698808146477888443887941193, 1.53726452050467517688175607532, 1.79808535618597171811345386664, 1.85330441331705195938803624285, 1.99448560552029089234674398879, 2.51128029913974098750449636693, 2.79326782988793334970850667669, 3.01426929548178601615416481343, 3.31605606642929933631129634443, 3.36009794106497061505609984199, 3.62120021017033561248461954104, 3.68291326029135179920936072981, 3.75376033726628338931156099432, 3.84967556953359069040535674931, 4.06613190895187954541176222906, 4.35946617545032578884051583535, 4.58938102270009655713289899411, 4.74141306071808377687636105008, 4.97817212847626108765272445169, 5.20283934819457970387306585801, 5.30188113725349336541977541325, 5.31165621003786517263243180217

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

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