Properties

Label 12-912e6-1.1-c0e6-0-2
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $0.00889020$
Root an. cond. $0.674646$
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  − 3·13-s + 3·19-s + 27-s + 3·43-s + 6·61-s + 3·67-s − 3·73-s − 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3·13-s + 3·19-s + 27-s + 3·43-s + 6·61-s + 3·67-s − 3·73-s − 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.00889020\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7566594888\)
\(L(\frac12)\) \(\approx\) \(0.7566594888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{3} + T^{6} \)
19 \( ( 1 - T + T^{2} )^{3} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )^{2} \)
37 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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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

−5.58203363676118635319975095362, −5.25422472975716030756694238752, −5.19957207990768060614414757973, −5.19202263432930426657095381939, −5.06941569722518177602294777764, −4.90754189264064532091556541060, −4.51270146605892299673393448812, −4.50620694711157630334599757815, −4.21334276750321749810035647422, −4.15852599832802457999956165105, −4.00247537330916088204788532662, −3.69656193780226036249956006023, −3.52335292919550285782092668387, −3.45249173387382402618476299658, −2.91483501260857518765720066120, −2.87578115028088518878629118637, −2.85137561760291189083279524415, −2.59443019736734478909813973360, −2.46609609339575195967818244721, −2.25367904216824911079160065794, −1.95280307646931823337428521543, −1.73102939741099338568308066525, −1.22796604635395947468663186980, −0.969150725369289885523021710750, −0.897879797936330743191016736577, 0.897879797936330743191016736577, 0.969150725369289885523021710750, 1.22796604635395947468663186980, 1.73102939741099338568308066525, 1.95280307646931823337428521543, 2.25367904216824911079160065794, 2.46609609339575195967818244721, 2.59443019736734478909813973360, 2.85137561760291189083279524415, 2.87578115028088518878629118637, 2.91483501260857518765720066120, 3.45249173387382402618476299658, 3.52335292919550285782092668387, 3.69656193780226036249956006023, 4.00247537330916088204788532662, 4.15852599832802457999956165105, 4.21334276750321749810035647422, 4.50620694711157630334599757815, 4.51270146605892299673393448812, 4.90754189264064532091556541060, 5.06941569722518177602294777764, 5.19202263432930426657095381939, 5.19957207990768060614414757973, 5.25422472975716030756694238752, 5.58203363676118635319975095362

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

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