Properties

Label 12-2808e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.902\times 10^{20}$
Sign $1$
Analytic cond. $7.57400$
Root an. cond. $1.18379$
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 − 2·8-s − 3·13-s − 9·16-s − 9·26-s + 3·31-s − 9·32-s − 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s + 9·124-s + 2·125-s + 127-s + 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3·2-s + 3·4-s − 2·8-s − 3·13-s − 9·16-s − 9·26-s + 3·31-s − 9·32-s − 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s + 9·124-s + 2·125-s + 127-s + 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{18} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(7.57400\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2808} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{18} \cdot 13^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1562342040\)
\(L(\frac12)\) \(\approx\) \(0.1562342040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.69239170164381780224682960116, −4.62159580546650659645236845382, −4.55237577424930014543191498964, −4.48762252774925482148859025575, −4.11736787987896011511390806745, −3.95956553388492202963162497915, −3.94078217396504192312297011477, −3.91216246553446994489744036426, −3.76700983914554452879312746881, −3.52888488343768563498869120316, −3.47914214937857586765163882822, −2.93528081555984216211104886028, −2.83908497302343051494416634656, −2.75557608694147840339422236317, −2.74442327573442324846179389723, −2.69058569325040982868417568263, −2.54509936796200466928039440813, −2.52771322891463967329343798997, −1.98950187653924777123905140316, −1.95686677742981913589531432341, −1.51662194028972419632152534636, −1.32895126404561611978888288606, −1.11340207452450151835295177409, −0.78353535400114509445401010587, −0.083638204825075081030009234183, 0.083638204825075081030009234183, 0.78353535400114509445401010587, 1.11340207452450151835295177409, 1.32895126404561611978888288606, 1.51662194028972419632152534636, 1.95686677742981913589531432341, 1.98950187653924777123905140316, 2.52771322891463967329343798997, 2.54509936796200466928039440813, 2.69058569325040982868417568263, 2.74442327573442324846179389723, 2.75557608694147840339422236317, 2.83908497302343051494416634656, 2.93528081555984216211104886028, 3.47914214937857586765163882822, 3.52888488343768563498869120316, 3.76700983914554452879312746881, 3.91216246553446994489744036426, 3.94078217396504192312297011477, 3.95956553388492202963162497915, 4.11736787987896011511390806745, 4.48762252774925482148859025575, 4.55237577424930014543191498964, 4.62159580546650659645236845382, 4.69239170164381780224682960116

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

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