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Properties

Label 12-3800e6-1.1-c0e6-0-3
Degree 1212
Conductor 3.011×10213.011\times 10^{21}
Sign 11
Analytic cond. 46.520446.5204
Root an. cond. 1.377111.37711
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 6·16-s − 6·19-s − 10·64-s + 18·76-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 3·4-s + 6·16-s − 6·19-s − 10·64-s + 18·76-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅512⋅1962^{18} \cdot 5^{12} \cdot 19^{6}
Sign: 11
Analytic conductor: 46.520446.5204
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅512⋅196, ( :[0]6), 1)(12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.43401605150.4340160515
L(12)L(\frac12) ≈\approx 0.43401605150.4340160515
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 (1+T2)3 ( 1 + T^{2} )^{3}
5 1 1
19 (1+T)6 ( 1 + T )^{6}
good3 1−T6+T12 1 - T^{6} + T^{12}
7 1−T6+T12 1 - T^{6} + T^{12}
11 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
13 1−T6+T12 1 - T^{6} + T^{12}
17 1−T6+T12 1 - T^{6} + T^{12}
23 1−T6+T12 1 - T^{6} + T^{12}
29 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
31 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
37 (1−T2+T4)3 ( 1 - T^{2} + T^{4} )^{3}
41 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
43 (1+T2)6 ( 1 + T^{2} )^{6}
47 (1−T2+T4)3 ( 1 - T^{2} + T^{4} )^{3}
53 1−T6+T12 1 - T^{6} + T^{12}
59 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
61 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
67 1−T6+T12 1 - T^{6} + T^{12}
71 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
73 1−T6+T12 1 - T^{6} + T^{12}
79 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
83 (1+T2)6 ( 1 + T^{2} )^{6}
89 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
97 (1+T2)6 ( 1 + T^{2} )^{6}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(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

−4.71758776344464674205871723641, −4.30327564757625793169181641224, −4.23774687609070138180374282642, −4.11195832858536234956969784778, −4.08343082045360354246643870477, −4.01147978866766937997893607457, −3.99365414189022273243799070410, −3.53530464658958635503126555797, −3.50683843871902413610775112285, −3.43340084325485757930990958980, −3.22465770066772047932746935620, −3.02063651519615724527946498287, −2.78718196518846251748011518083, −2.76541946939768003642827771560, −2.35101914896618355435565505983, −2.28381096615094497585937348680, −2.21520706055977960957688491061, −1.95578864965617538534759869238, −1.70478601775394990787670295736, −1.68525463392386657561590875773, −1.31404186641320955428549139283, −1.29774912106516932850348464564, −0.59391308421665691461222734084, −0.56370632011863495275363234968, −0.42637819116909785239484517734, 0.42637819116909785239484517734, 0.56370632011863495275363234968, 0.59391308421665691461222734084, 1.29774912106516932850348464564, 1.31404186641320955428549139283, 1.68525463392386657561590875773, 1.70478601775394990787670295736, 1.95578864965617538534759869238, 2.21520706055977960957688491061, 2.28381096615094497585937348680, 2.35101914896618355435565505983, 2.76541946939768003642827771560, 2.78718196518846251748011518083, 3.02063651519615724527946498287, 3.22465770066772047932746935620, 3.43340084325485757930990958980, 3.50683843871902413610775112285, 3.53530464658958635503126555797, 3.99365414189022273243799070410, 4.01147978866766937997893607457, 4.08343082045360354246643870477, 4.11195832858536234956969784778, 4.23774687609070138180374282642, 4.30327564757625793169181641224, 4.71758776344464674205871723641

Graph of the ZZ-function along the critical line

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