Properties

Label 12-1183e6-1.1-c0e6-0-0
Degree $12$
Conductor $2.741\times 10^{18}$
Sign $1$
Analytic cond. $0.0423497$
Root an. cond. $0.768370$
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  + 2-s + 4-s − 3·7-s − 3·9-s + 11-s − 3·14-s − 3·18-s + 22-s + 23-s + 6·25-s − 3·28-s + 29-s − 3·36-s + 37-s + 43-s + 44-s + 46-s + 3·49-s + 6·50-s − 2·53-s + 58-s + 9·63-s + 67-s + 71-s + 74-s − 3·77-s − 2·79-s + ⋯
L(s)  = 1  + 2-s + 4-s − 3·7-s − 3·9-s + 11-s − 3·14-s − 3·18-s + 22-s + 23-s + 6·25-s − 3·28-s + 29-s − 3·36-s + 37-s + 43-s + 44-s + 46-s + 3·49-s + 6·50-s − 2·53-s + 58-s + 9·63-s + 67-s + 71-s + 74-s − 3·77-s − 2·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.0423497\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1183} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 13^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004887484\)
\(L(\frac12)\) \(\approx\) \(1.004887484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T + T^{2} )^{3} \)
13 \( 1 \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
3 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 - T )^{6}( 1 + T )^{6} \)
11 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
47 \( ( 1 - T )^{6}( 1 + T )^{6} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
71 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−5.36917023065666130134037465954, −5.09246166150546607251209583061, −5.08854729075310595449900778446, −4.91082963089139420612313416087, −4.81967325135694429343913774321, −4.77977803047232652221772249196, −4.28456390588928114035296435740, −4.26760307831401861117243192288, −4.20389513877995572958941862345, −3.75403825897156642798294605088, −3.49591029498450966710325083184, −3.46731441658396192930491545863, −3.30633538852180643121480777003, −3.18865786731047742011701616473, −3.00084110576151595342055943756, −2.91549534127026354251162891927, −2.71700544961423916570107770985, −2.62741868770318602662041303307, −2.62450773620482330812293097375, −2.42115420017392378866039383174, −1.82360317619439863923624021570, −1.55758082602634536846886157221, −1.18369570412579761504192909885, −0.882784683440378454382926322735, −0.67460351321824723772964645058, 0.67460351321824723772964645058, 0.882784683440378454382926322735, 1.18369570412579761504192909885, 1.55758082602634536846886157221, 1.82360317619439863923624021570, 2.42115420017392378866039383174, 2.62450773620482330812293097375, 2.62741868770318602662041303307, 2.71700544961423916570107770985, 2.91549534127026354251162891927, 3.00084110576151595342055943756, 3.18865786731047742011701616473, 3.30633538852180643121480777003, 3.46731441658396192930491545863, 3.49591029498450966710325083184, 3.75403825897156642798294605088, 4.20389513877995572958941862345, 4.26760307831401861117243192288, 4.28456390588928114035296435740, 4.77977803047232652221772249196, 4.81967325135694429343913774321, 4.91082963089139420612313416087, 5.08854729075310595449900778446, 5.09246166150546607251209583061, 5.36917023065666130134037465954

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

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