Properties

Label 12-2888e6-1.1-c0e6-0-2
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
Motivic weight $0$
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  + 3·7-s + 8-s + 2·27-s − 12·37-s + 6·49-s + 3·56-s − 3·107-s − 3·121-s + ⋯
L(s)  = 1  + 3·7-s + 8-s + 2·27-s − 12·37-s + 6·49-s + 3·56-s − 3·107-s − 3·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657243111\)
\(L(\frac12)\) \(\approx\) \(1.657243111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( ( 1 - T^{3} + T^{6} )^{2} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T^{3} + T^{6} )^{2} \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 + T )^{12} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T^{3} + T^{6} )^{2} \)
59 \( ( 1 - T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−4.86410411817021912244948914985, −4.67363137297238249845688515887, −4.60737062821258431640247222085, −4.39969466412613144149664653094, −4.06317884006880503976710289304, −3.98551967198951257638640478051, −3.80226615829160823020501093420, −3.70344844579570696876026231640, −3.69305731388831612042068609434, −3.68248748438174691994156191238, −3.10786230711116819668335042260, −3.06206898268024071980378589579, −2.96246869729841012150783096854, −2.87267701004437449148950403320, −2.64689493312049062090468363354, −2.13793859830913478302467335235, −2.12597621974429726118213965219, −1.92355779555376609970812347106, −1.89787353921309520428918247805, −1.62497127028816034218825278292, −1.59592751060580728261813543724, −1.44297436389906666318456100720, −1.12403190989088580097068191227, −1.07024134356112830537930207254, −0.34477159664437119229057282217, 0.34477159664437119229057282217, 1.07024134356112830537930207254, 1.12403190989088580097068191227, 1.44297436389906666318456100720, 1.59592751060580728261813543724, 1.62497127028816034218825278292, 1.89787353921309520428918247805, 1.92355779555376609970812347106, 2.12597621974429726118213965219, 2.13793859830913478302467335235, 2.64689493312049062090468363354, 2.87267701004437449148950403320, 2.96246869729841012150783096854, 3.06206898268024071980378589579, 3.10786230711116819668335042260, 3.68248748438174691994156191238, 3.69305731388831612042068609434, 3.70344844579570696876026231640, 3.80226615829160823020501093420, 3.98551967198951257638640478051, 4.06317884006880503976710289304, 4.39969466412613144149664653094, 4.60737062821258431640247222085, 4.67363137297238249845688515887, 4.86410411817021912244948914985

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

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