Properties

Label 12-912e6-1.1-c1e6-0-1
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
Motivic weight $1$
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  + 21·13-s − 21·19-s + 9·27-s − 15·43-s − 42·61-s − 15·67-s − 21·73-s − 12·79-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 255·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 5.82·13-s − 4.81·19-s + 1.73·27-s − 2.28·43-s − 5.37·61-s − 1.83·67-s − 2.45·73-s − 1.35·79-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 19.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.206231248\)
\(L(\frac12)\) \(\approx\) \(1.206231248\)
\(L(\frac{3}{2})\) 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 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
19 \( ( 1 + 7 T + p T^{2} )^{3} \)
good5 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
7 \( ( 1 + 37 T^{3} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \)
13 \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 19 T^{3} + p^{3} T^{6} ) \)
17 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
23 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
29 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
31 \( ( 1 - 289 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \)
37 \( ( 1 - 433 T^{3} + p^{3} T^{6} )( 1 + 433 T^{3} + p^{3} T^{6} ) \)
41 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
43 \( ( 1 + 5 T + p T^{2} )^{3}( 1 + 71 T^{3} + p^{3} T^{6} ) \)
47 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
53 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
59 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
61 \( ( 1 + 14 T + p T^{2} )^{3}( 1 + 719 T^{3} + p^{3} T^{6} ) \)
67 \( ( 1 + 5 T + p T^{2} )^{3}( 1 - 1007 T^{3} + p^{3} T^{6} ) \)
71 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
73 \( ( 1 + 7 T + p T^{2} )^{3}( 1 + 271 T^{3} + p^{3} T^{6} ) \)
79 \( ( 1 + 4 T + p T^{2} )^{3}( 1 - 1387 T^{3} + p^{3} T^{6} ) \)
83 \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \)
89 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
97 \( ( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 - 523 T^{3} + p^{3} T^{6} ) \)
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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

−5.47731227416401486833575471745, −5.37398707399597854418993284909, −4.94067266813186149999322670137, −4.63327425865627111289550933589, −4.53884713485692896434781642291, −4.48773198333909638416786661291, −4.29959051888338104138390885328, −4.26352403436147591425285144111, −4.19540971457352515730002880485, −3.73962732026927996028591217925, −3.60153544187186135268942487401, −3.48397751959210105235492987933, −3.47045655995572248456979927091, −3.01483976417908673992773960655, −2.98888979795206042521358639556, −2.72172734407516979288358045408, −2.68954929573846037864067504138, −2.17226830798762035787939270521, −1.70068657890862199543314858808, −1.69026201935659860480343080587, −1.65550549689846848386071716949, −1.49312232730551980750133389657, −1.15692455082068337992660149074, −0.72573505280979590223468713872, −0.17423649936896607530206368046, 0.17423649936896607530206368046, 0.72573505280979590223468713872, 1.15692455082068337992660149074, 1.49312232730551980750133389657, 1.65550549689846848386071716949, 1.69026201935659860480343080587, 1.70068657890862199543314858808, 2.17226830798762035787939270521, 2.68954929573846037864067504138, 2.72172734407516979288358045408, 2.98888979795206042521358639556, 3.01483976417908673992773960655, 3.47045655995572248456979927091, 3.48397751959210105235492987933, 3.60153544187186135268942487401, 3.73962732026927996028591217925, 4.19540971457352515730002880485, 4.26352403436147591425285144111, 4.29959051888338104138390885328, 4.48773198333909638416786661291, 4.53884713485692896434781642291, 4.63327425865627111289550933589, 4.94067266813186149999322670137, 5.37398707399597854418993284909, 5.47731227416401486833575471745

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

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