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Properties

Label 12-285e6-1.1-c0e6-0-1
Degree $12$
Conductor $5.359\times 10^{14}$
Sign $1$
Analytic cond. $8.27964\times 10^{-6}$
Root an. cond. $0.377138$
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  + 3·2-s + 3·4-s − 8-s − 6·16-s − 6·17-s + 27-s − 6·32-s − 18·34-s + 3·47-s − 3·49-s + 3·53-s + 3·54-s + 64-s − 18·68-s + 9·94-s − 9·98-s + 9·106-s + 3·108-s − 3·109-s − 3·121-s + 125-s + 127-s + 9·128-s + 131-s + 6·136-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3·2-s + 3·4-s − 8-s − 6·16-s − 6·17-s + 27-s − 6·32-s − 18·34-s + 3·47-s − 3·49-s + 3·53-s + 3·54-s + 64-s − 18·68-s + 9·94-s − 9·98-s + 9·106-s + 3·108-s − 3·109-s − 3·121-s + 125-s + 127-s + 9·128-s + 131-s + 6·136-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8.27964\times 10^{-6}\)
Root analytic conductor: \(0.377138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5189401395\)
\(L(\frac12)\) \(\approx\) \(0.5189401395\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( 1 + T^{3} + T^{6} \)
good2 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
23 \( ( 1 - T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
53 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
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

−6.55279198210966454229377746983, −6.45494964737171486888241368266, −6.30895244133513546097602572285, −6.20547253459175653183572755737, −6.11329808608514073725027894814, −5.54275312247184961434295165470, −5.49348741541786826041408957845, −5.34548096444707007130824244582, −5.06412496625005352757204882521, −5.02055399347997694546157077876, −4.66319841044178997023146095721, −4.57010303344770628017073012941, −4.54861483542438634931694193169, −4.18344275264860352061814683263, −4.04411960311907984571054516076, −3.99394552226346928766970156371, −3.81251638633129187039886799262, −3.61754761610011142657138729550, −3.15058520749016528264260621096, −2.71484977217962682589049111425, −2.60031055058779045732555514958, −2.47248564917661304286165415656, −2.39672727532387591428564775451, −1.96321002090837839465893124632, −1.38160686124600738178473975199, 1.38160686124600738178473975199, 1.96321002090837839465893124632, 2.39672727532387591428564775451, 2.47248564917661304286165415656, 2.60031055058779045732555514958, 2.71484977217962682589049111425, 3.15058520749016528264260621096, 3.61754761610011142657138729550, 3.81251638633129187039886799262, 3.99394552226346928766970156371, 4.04411960311907984571054516076, 4.18344275264860352061814683263, 4.54861483542438634931694193169, 4.57010303344770628017073012941, 4.66319841044178997023146095721, 5.02055399347997694546157077876, 5.06412496625005352757204882521, 5.34548096444707007130824244582, 5.49348741541786826041408957845, 5.54275312247184961434295165470, 6.11329808608514073725027894814, 6.20547253459175653183572755737, 6.30895244133513546097602572285, 6.45494964737171486888241368266, 6.55279198210966454229377746983

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

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