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Properties

Label 12-2736e6-1.1-c0e6-0-2
Degree $12$
Conductor $4.195\times 10^{20}$
Sign $1$
Analytic cond. $6.48095$
Root an. cond. $1.16852$
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 + 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 + 223-s + ⋯
L(s)  = 1  + 3·13-s + 3·19-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 + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(6.48095\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.556288068\)
\(L(\frac12)\) \(\approx\) \(3.556288068\)
\(L(1)\) 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 \( 1 \)
19 \( ( 1 - T + T^{2} )^{3} \)
good5 \( 1 - T^{6} + T^{12} \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
17 \( 1 - T^{6} + T^{12} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( 1 - T^{6} + T^{12} \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( 1 - T^{6} + T^{12} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( 1 - T^{6} + T^{12} \)
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 \( 1 - T^{6} + T^{12} \)
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

−5.01750793759783187173446403523, −4.67026880240693195162183169745, −4.29419107931578137219567759459, −4.17792939747755675968107813779, −4.12869817702308375810093684644, −4.11859874466801241384100931819, −4.02607581809325126458396775564, −3.50761405254607011434148827038, −3.48997039377118444582394382187, −3.42187450396559928580983996589, −3.35347556209531539868810905284, −3.33577941178577100644431955216, −3.25943900846408191559638608243, −2.62515174892172630128278269246, −2.57761491931209092971974309877, −2.34519510248242047685385955893, −2.33910994776101595092448930334, −2.18261228895643057486229214107, −2.04753858091819288430427177756, −1.65383534467245632856282335117, −1.27922242948173398644991721308, −1.11395685805256252589687116167, −1.02134320580787487600268775313, −0.946199370142556663411489282118, −0.836938239494087750550743392080, 0.836938239494087750550743392080, 0.946199370142556663411489282118, 1.02134320580787487600268775313, 1.11395685805256252589687116167, 1.27922242948173398644991721308, 1.65383534467245632856282335117, 2.04753858091819288430427177756, 2.18261228895643057486229214107, 2.33910994776101595092448930334, 2.34519510248242047685385955893, 2.57761491931209092971974309877, 2.62515174892172630128278269246, 3.25943900846408191559638608243, 3.33577941178577100644431955216, 3.35347556209531539868810905284, 3.42187450396559928580983996589, 3.48997039377118444582394382187, 3.50761405254607011434148827038, 4.02607581809325126458396775564, 4.11859874466801241384100931819, 4.12869817702308375810093684644, 4.17792939747755675968107813779, 4.29419107931578137219567759459, 4.67026880240693195162183169745, 5.01750793759783187173446403523

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

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