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Properties

Label 12-1944e6-1.1-c1e6-0-2
Degree $12$
Conductor $5.397\times 10^{19}$
Sign $1$
Analytic cond. $1.39907\times 10^{7}$
Root an. cond. $3.93991$
Motivic weight $1$
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  + 3·7-s + 3·11-s + 3·13-s + 18·17-s − 18·19-s + 6·23-s + 12·25-s − 6·29-s + 6·31-s − 30·37-s − 3·41-s + 6·47-s + 21·49-s − 12·53-s + 3·59-s + 18·61-s + 24·67-s − 30·71-s − 12·73-s + 9·77-s + 12·79-s + 18·83-s + 36·89-s + 9·91-s + 9·97-s + 3·103-s + 6·107-s + ⋯
L(s)  = 1  + 1.13·7-s + 0.904·11-s + 0.832·13-s + 4.36·17-s − 4.12·19-s + 1.25·23-s + 12/5·25-s − 1.11·29-s + 1.07·31-s − 4.93·37-s − 0.468·41-s + 0.875·47-s + 3·49-s − 1.64·53-s + 0.390·59-s + 2.30·61-s + 2.93·67-s − 3.56·71-s − 1.40·73-s + 1.02·77-s + 1.35·79-s + 1.97·83-s + 3.81·89-s + 0.943·91-s + 0.913·97-s + 0.295·103-s + 0.580·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{30}\)
Sign: $1$
Analytic conductor: \(1.39907\times 10^{7}\)
Root analytic conductor: \(3.93991\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{30} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.35022317\)
\(L(\frac12)\) \(\approx\) \(14.35022317\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 12 T^{2} + 2 T^{3} + 84 T^{4} - 12 T^{5} - 469 T^{6} - 12 p T^{7} + 84 p^{2} T^{8} + 2 p^{3} T^{9} - 12 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 3 T - 12 T^{2} + 15 T^{3} + 177 T^{4} - 12 p T^{5} - 1321 T^{6} - 12 p^{2} T^{7} + 177 p^{2} T^{8} + 15 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T - 18 T^{2} + 53 T^{3} + 21 p T^{4} - 390 T^{5} - 1957 T^{6} - 390 p T^{7} + 21 p^{3} T^{8} + 53 p^{3} T^{9} - 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T - 12 T^{2} + 19 T^{3} + 45 T^{4} + 270 T^{5} - 1179 T^{6} + 270 p T^{7} + 45 p^{2} T^{8} + 19 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 9 T + 66 T^{2} - p^{2} T^{3} + 66 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 9 T + 63 T^{2} + 289 T^{3} + 63 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 6 T - 24 T^{2} + 50 T^{3} + 990 T^{4} + 1086 T^{5} - 35485 T^{6} + 1086 p T^{7} + 990 p^{2} T^{8} + 50 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T - 60 T^{2} - 126 T^{3} + 4326 T^{4} + 5334 T^{5} - 125561 T^{6} + 5334 p T^{7} + 4326 p^{2} T^{8} - 126 p^{3} T^{9} - 60 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 6 T - 48 T^{2} + 98 T^{3} + 2646 T^{4} + 198 T^{5} - 103149 T^{6} + 198 p T^{7} + 2646 p^{2} T^{8} + 98 p^{3} T^{9} - 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 15 T + 147 T^{2} + 951 T^{3} + 147 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 3 T - 60 T^{2} - 387 T^{3} + 1041 T^{4} + 8562 T^{5} + 15721 T^{6} + 8562 p T^{7} + 1041 p^{2} T^{8} - 387 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 54 T^{2} + 250 T^{3} + 594 T^{4} - 6750 T^{5} + 49251 T^{6} - 6750 p T^{7} + 594 p^{2} T^{8} + 250 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 6 T - 114 T^{2} + 234 T^{3} + 12048 T^{4} - 13704 T^{5} - 607889 T^{6} - 13704 p T^{7} + 12048 p^{2} T^{8} + 234 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 114 T^{2} + 367 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 108 T^{2} - 109 T^{3} + 6423 T^{4} + 16818 T^{5} - 413581 T^{6} + 16818 p T^{7} + 6423 p^{2} T^{8} - 109 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 72 T^{2} - 218 T^{3} + 9270 T^{4} - 57654 T^{5} + 35895 T^{6} - 57654 p T^{7} + 9270 p^{2} T^{8} - 218 p^{3} T^{9} + 72 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 24 T + 204 T^{2} - 1882 T^{3} + 28908 T^{4} - 221940 T^{5} + 1155663 T^{6} - 221940 p T^{7} + 28908 p^{2} T^{8} - 1882 p^{3} T^{9} + 204 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 15 T + 171 T^{2} + 1193 T^{3} + 171 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 6 T + 150 T^{2} + 965 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 12 T - 30 T^{2} + 298 T^{3} + 2214 T^{4} + 47430 T^{5} - 881181 T^{6} + 47430 p T^{7} + 2214 p^{2} T^{8} + 298 p^{3} T^{9} - 30 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 186 T^{2} - 542 T^{3} - 9888 T^{4} + 154344 T^{5} - 1726273 T^{6} + 154344 p T^{7} - 9888 p^{2} T^{8} - 542 p^{3} T^{9} + 186 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 18 T + 258 T^{2} - 2565 T^{3} + 258 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 9 T - 45 T^{2} + 236 T^{3} - 2511 T^{4} + 49005 T^{5} - 7530 T^{6} + 49005 p T^{7} - 2511 p^{2} T^{8} + 236 p^{3} T^{9} - 45 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.78815581981614956560428538361, −4.57584806665705344858322905773, −4.48060413309100986455718078476, −4.37408456018876162416901611019, −4.22081119862151708941641893724, −3.90151238655244560973281640601, −3.87327577235197191543393098866, −3.78084486484171726830578415357, −3.49593693349935609814736272317, −3.33946593586858521823034095098, −3.25282834256115731842073174554, −3.16763140533063905078508976757, −3.10767390946305088494459687531, −2.53951957528363474800658671607, −2.49120126521902928616892242372, −2.40252496750302308512628914106, −2.10799562586097935418377140319, −1.74973798022909890611511245113, −1.74501040200199628086014825078, −1.51715998838072664788296455895, −1.50721854241807753178441982535, −1.00433889772730981903142064994, −0.868598164027496787764270341810, −0.61575327861448838687141969137, −0.46050891259060072117945698069, 0.46050891259060072117945698069, 0.61575327861448838687141969137, 0.868598164027496787764270341810, 1.00433889772730981903142064994, 1.50721854241807753178441982535, 1.51715998838072664788296455895, 1.74501040200199628086014825078, 1.74973798022909890611511245113, 2.10799562586097935418377140319, 2.40252496750302308512628914106, 2.49120126521902928616892242372, 2.53951957528363474800658671607, 3.10767390946305088494459687531, 3.16763140533063905078508976757, 3.25282834256115731842073174554, 3.33946593586858521823034095098, 3.49593693349935609814736272317, 3.78084486484171726830578415357, 3.87327577235197191543393098866, 3.90151238655244560973281640601, 4.22081119862151708941641893724, 4.37408456018876162416901611019, 4.48060413309100986455718078476, 4.57584806665705344858322905773, 4.78815581981614956560428538361

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

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