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Properties

Label 12-3479e6-1.1-c0e6-0-3
Degree $12$
Conductor $1.773\times 10^{21}$
Sign $1$
Analytic cond. $27.3948$
Root an. cond. $1.31766$
Motivic weight $0$
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 + 3-s + 4-s + 5-s + 6-s + 9-s + 10-s + 12-s + 15-s + 18-s + 19-s + 20-s + 25-s − 2·29-s + 30-s + 36-s + 37-s + 38-s − 2·43-s + 45-s + 50-s + 57-s − 2·58-s + 60-s + 6·71-s + 73-s + 74-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 9-s + 10-s + 12-s + 15-s + 18-s + 19-s + 20-s + 25-s − 2·29-s + 30-s + 36-s + 37-s + 38-s − 2·43-s + 45-s + 50-s + 57-s − 2·58-s + 60-s + 6·71-s + 73-s + 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 71^{6}\)
Sign: $1$
Analytic conductor: \(27.3948\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 71^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.609464111\)
\(L(\frac12)\) \(\approx\) \(8.609464111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( ( 1 - T )^{6} \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
5 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T )^{6}( 1 + T )^{6} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
79 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
89 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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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

−4.71704183775806056314363994881, −4.51973035955392498793463074680, −4.25067760423411041997258828861, −4.07739474649652123509236687199, −3.87554484988257245755131834149, −3.86896851509917259641837419447, −3.77128071265738050502252991390, −3.69986542689964664009349716993, −3.61715431205407567038751244050, −3.21980219694955352643043937979, −3.03948837158146282785114599234, −3.01985465048293395942077836386, −2.99641673417018749870916723505, −2.68910684223678675235976638759, −2.57869757947089792291604471301, −2.37721643520263157803459343976, −2.28724303925064485203233853192, −1.97742482905351543108478641403, −1.82308950240811652506616147193, −1.73932495653428754563371129909, −1.69202617555782880892650788483, −1.42180460640654870876114204339, −0.975928303322383927875439227712, −0.897539491788389904159860570740, −0.60028339318405292262407915233, 0.60028339318405292262407915233, 0.897539491788389904159860570740, 0.975928303322383927875439227712, 1.42180460640654870876114204339, 1.69202617555782880892650788483, 1.73932495653428754563371129909, 1.82308950240811652506616147193, 1.97742482905351543108478641403, 2.28724303925064485203233853192, 2.37721643520263157803459343976, 2.57869757947089792291604471301, 2.68910684223678675235976638759, 2.99641673417018749870916723505, 3.01985465048293395942077836386, 3.03948837158146282785114599234, 3.21980219694955352643043937979, 3.61715431205407567038751244050, 3.69986542689964664009349716993, 3.77128071265738050502252991390, 3.86896851509917259641837419447, 3.87554484988257245755131834149, 4.07739474649652123509236687199, 4.25067760423411041997258828861, 4.51973035955392498793463074680, 4.71704183775806056314363994881

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

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