Properties

Label 12-2888e6-1.1-c0e6-0-3
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s − 9·16-s + 3·17-s − 3·25-s − 2·27-s + 9·32-s − 9·34-s + 3·43-s + 6·49-s + 9·50-s + 6·54-s + 3·64-s + 9·68-s − 9·86-s + 3·89-s − 18·98-s − 9·100-s − 6·107-s − 6·108-s + 127-s − 18·128-s + 131-s + 6·136-s + 137-s + 139-s + ⋯
L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s − 9·16-s + 3·17-s − 3·25-s − 2·27-s + 9·32-s − 9·34-s + 3·43-s + 6·49-s + 9·50-s + 6·54-s + 3·64-s + 9·68-s − 9·86-s + 3·89-s − 18·98-s − 9·100-s − 6·107-s − 6·108-s + 127-s − 18·128-s + 131-s + 6·136-s + 137-s + 139-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: induced by $\chi_{2888} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2022322870\)
\(L(\frac12)\) \(\approx\) \(0.2022322870\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.71816454638715442129940559056, −4.58768929848599925020705763823, −4.42397315127002355419516279147, −4.03723646811873972787289549516, −4.03402513160551732087726176585, −3.95307650904102647586440589977, −3.94815570727009404188752212725, −3.85757521192815391971078139964, −3.71863990237356771452906315787, −3.59732534699859407151847611648, −3.21606771023013407189258209557, −2.85177531042252682158178136985, −2.80825583896444296002341311009, −2.76738339141353640070273612832, −2.41194007802340266673115447423, −2.28733973991170197725997292149, −2.07592113183844305123237163760, −1.82621902203459743614142187765, −1.78165928957934329060036778495, −1.74901778776636730931783788250, −1.09733123894287451397950762083, −1.07526277125942176532524721440, −1.05357064850437119726910331811, −0.810469590793909577491085084232, −0.36831052152737027040768285476, 0.36831052152737027040768285476, 0.810469590793909577491085084232, 1.05357064850437119726910331811, 1.07526277125942176532524721440, 1.09733123894287451397950762083, 1.74901778776636730931783788250, 1.78165928957934329060036778495, 1.82621902203459743614142187765, 2.07592113183844305123237163760, 2.28733973991170197725997292149, 2.41194007802340266673115447423, 2.76738339141353640070273612832, 2.80825583896444296002341311009, 2.85177531042252682158178136985, 3.21606771023013407189258209557, 3.59732534699859407151847611648, 3.71863990237356771452906315787, 3.85757521192815391971078139964, 3.94815570727009404188752212725, 3.95307650904102647586440589977, 4.03402513160551732087726176585, 4.03723646811873972787289549516, 4.42397315127002355419516279147, 4.58768929848599925020705763823, 4.71816454638715442129940559056

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

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