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Properties

Label 12-5415e6-1.1-c1e6-0-6
Degree $12$
Conductor $2.521\times 10^{22}$
Sign $1$
Analytic cond. $6.53511\times 10^{9}$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 6·3-s − 4·4-s + 6·5-s + 6·6-s + 2·7-s + 5·8-s + 21·9-s − 6·10-s + 6·11-s + 24·12-s − 4·13-s − 2·14-s − 36·15-s + 5·16-s + 17-s − 21·18-s − 24·20-s − 12·21-s − 6·22-s + 3·23-s − 30·24-s + 21·25-s + 4·26-s − 56·27-s − 8·28-s − 27·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 3.46·3-s − 2·4-s + 2.68·5-s + 2.44·6-s + 0.755·7-s + 1.76·8-s + 7·9-s − 1.89·10-s + 1.80·11-s + 6.92·12-s − 1.10·13-s − 0.534·14-s − 9.29·15-s + 5/4·16-s + 0.242·17-s − 4.94·18-s − 5.36·20-s − 2.61·21-s − 1.27·22-s + 0.625·23-s − 6.12·24-s + 21/5·25-s + 0.784·26-s − 10.7·27-s − 1.51·28-s − 5.01·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.53511\times 10^{9}\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 - T )^{6} \)
19 \( 1 \)
good2 \( 1 + T + 5 T^{2} + p^{2} T^{3} + 7 p T^{4} + 13 T^{5} + 33 T^{6} + 13 p T^{7} + 7 p^{3} T^{8} + p^{5} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 22 T^{2} - 37 T^{3} + 247 T^{4} - 363 T^{5} + 1909 T^{6} - 363 p T^{7} + 247 p^{2} T^{8} - 37 p^{3} T^{9} + 22 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 71 T^{2} - 317 T^{3} + 2031 T^{4} - 6838 T^{5} + 30161 T^{6} - 6838 p T^{7} + 2031 p^{2} T^{8} - 317 p^{3} T^{9} + 71 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 47 T^{2} + 164 T^{3} + 1246 T^{4} + 3556 T^{5} + 19843 T^{6} + 3556 p T^{7} + 1246 p^{2} T^{8} + 164 p^{3} T^{9} + 47 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - T + 56 T^{2} + 50 T^{3} + 1332 T^{4} + 3245 T^{5} + 23371 T^{6} + 3245 p T^{7} + 1332 p^{2} T^{8} + 50 p^{3} T^{9} + 56 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T + 120 T^{2} - 268 T^{3} + 6224 T^{4} - 10639 T^{5} + 183767 T^{6} - 10639 p T^{7} + 6224 p^{2} T^{8} - 268 p^{3} T^{9} + 120 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 27 T + 369 T^{2} + 3519 T^{3} + 27614 T^{4} + 6512 p T^{5} + 1108087 T^{6} + 6512 p^{2} T^{7} + 27614 p^{2} T^{8} + 3519 p^{3} T^{9} + 369 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 17 T + 290 T^{2} + 2872 T^{3} + 27206 T^{4} + 184197 T^{5} + 1187653 T^{6} + 184197 p T^{7} + 27206 p^{2} T^{8} + 2872 p^{3} T^{9} + 290 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 17 T + 244 T^{2} + 2265 T^{3} + 19019 T^{4} + 128252 T^{5} + 839929 T^{6} + 128252 p T^{7} + 19019 p^{2} T^{8} + 2265 p^{3} T^{9} + 244 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 11 T + 206 T^{2} + 1507 T^{3} + 17121 T^{4} + 96008 T^{5} + 852751 T^{6} + 96008 p T^{7} + 17121 p^{2} T^{8} + 1507 p^{3} T^{9} + 206 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 119 T^{2} - 36 T^{3} + 7867 T^{4} - 142 p T^{5} + 467515 T^{6} - 142 p^{2} T^{7} + 7867 p^{2} T^{8} - 36 p^{3} T^{9} + 119 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 252 T^{2} + 2224 T^{3} + 23543 T^{4} + 156685 T^{5} + 1312363 T^{6} + 156685 p T^{7} + 23543 p^{2} T^{8} + 2224 p^{3} T^{9} + 252 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 14 T + 304 T^{2} + 2866 T^{3} + 36002 T^{4} + 256106 T^{5} + 2405915 T^{6} + 256106 p T^{7} + 36002 p^{2} T^{8} + 2866 p^{3} T^{9} + 304 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 19 T + 341 T^{2} + 3401 T^{3} + 33868 T^{4} + 233532 T^{5} + 2003595 T^{6} + 233532 p T^{7} + 33868 p^{2} T^{8} + 3401 p^{3} T^{9} + 341 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 16 T + 201 T^{2} + 2476 T^{3} + 23038 T^{4} + 201637 T^{5} + 28887 p T^{6} + 201637 p T^{7} + 23038 p^{2} T^{8} + 2476 p^{3} T^{9} + 201 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T + 97 T^{2} + 765 T^{3} + 13500 T^{4} + 73268 T^{5} + 865879 T^{6} + 73268 p T^{7} + 13500 p^{2} T^{8} + 765 p^{3} T^{9} + 97 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 3 T + 96 T^{2} + 308 T^{3} + 15697 T^{4} + 32588 T^{5} + 870645 T^{6} + 32588 p T^{7} + 15697 p^{2} T^{8} + 308 p^{3} T^{9} + 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 283 T^{2} - 1672 T^{3} + 39278 T^{4} - 193725 T^{5} + 3535757 T^{6} - 193725 p T^{7} + 39278 p^{2} T^{8} - 1672 p^{3} T^{9} + 283 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 19 T + 464 T^{2} + 6077 T^{3} + 88949 T^{4} + 876594 T^{5} + 9255517 T^{6} + 876594 p T^{7} + 88949 p^{2} T^{8} + 6077 p^{3} T^{9} + 464 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + T - 8 T^{2} - 89 T^{3} + 5721 T^{4} - 27746 T^{5} - 637987 T^{6} - 27746 p T^{7} + 5721 p^{2} T^{8} - 89 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 31 T + 739 T^{2} + 11393 T^{3} + 154284 T^{4} + 1653226 T^{5} + 16963877 T^{6} + 1653226 p T^{7} + 154284 p^{2} T^{8} + 11393 p^{3} T^{9} + 739 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T + 64 T^{2} + 50 T^{3} + 6544 T^{4} + 110607 T^{5} + 1586929 T^{6} + 110607 p T^{7} + 6544 p^{2} T^{8} + 50 p^{3} T^{9} + 64 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.72365388844566205167695525456, −4.59306497447051441865263202911, −4.37176444108934964509266536513, −4.35497343287687857534133715597, −4.19445010639142005258305125280, −3.92927268363955886734227209493, −3.91854118461553494523780924517, −3.68620405016997928904284306119, −3.50682515651367408432699341728, −3.37323874039482775689049104091, −3.25400792538181645953232101023, −3.24526083569118124457803450458, −3.04448537415985958224296424724, −2.78580252650158426744786814707, −2.34738939277272029705820861219, −2.08596870746026329146102413961, −2.04420509695208230122502334044, −1.95448254932807628355568737832, −1.77075449575766022598261504727, −1.61978616530591328099823974793, −1.58022737219206378307836968374, −1.39669453405522384666510161453, −1.22044078080226967836787065799, −1.17061427793179570697263895504, −1.01506174203872736327517266103, 0, 0, 0, 0, 0, 0, 1.01506174203872736327517266103, 1.17061427793179570697263895504, 1.22044078080226967836787065799, 1.39669453405522384666510161453, 1.58022737219206378307836968374, 1.61978616530591328099823974793, 1.77075449575766022598261504727, 1.95448254932807628355568737832, 2.04420509695208230122502334044, 2.08596870746026329146102413961, 2.34738939277272029705820861219, 2.78580252650158426744786814707, 3.04448537415985958224296424724, 3.24526083569118124457803450458, 3.25400792538181645953232101023, 3.37323874039482775689049104091, 3.50682515651367408432699341728, 3.68620405016997928904284306119, 3.91854118461553494523780924517, 3.92927268363955886734227209493, 4.19445010639142005258305125280, 4.35497343287687857534133715597, 4.37176444108934964509266536513, 4.59306497447051441865263202911, 4.72365388844566205167695525456

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

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