Properties

Label 12-912e6-1.1-c1e6-0-13
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s − 3·7-s − 3·13-s − 3·17-s + 18·19-s + 21·23-s + 45·25-s − 27-s − 3·29-s − 9·31-s − 27·35-s − 18·37-s − 15·41-s − 3·43-s + 9·47-s + 21·49-s + 12·53-s − 27·59-s + 3·61-s − 27·65-s − 21·67-s − 39·71-s + 36·73-s + 45·79-s + 27·83-s − 27·85-s − 30·89-s + ⋯
L(s)  = 1  + 4.02·5-s − 1.13·7-s − 0.832·13-s − 0.727·17-s + 4.12·19-s + 4.37·23-s + 9·25-s − 0.192·27-s − 0.557·29-s − 1.61·31-s − 4.56·35-s − 2.95·37-s − 2.34·41-s − 0.457·43-s + 1.31·47-s + 3·49-s + 1.64·53-s − 3.51·59-s + 0.384·61-s − 3.34·65-s − 2.56·67-s − 4.62·71-s + 4.21·73-s + 5.06·79-s + 2.96·83-s − 2.92·85-s − 3.17·89-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

\(L(1)\) \(\approx\) \(11.40901272\)
\(L(\frac12)\) \(\approx\) \(11.40901272\)
\(L(\frac{3}{2})\) 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 \( 1 + T^{3} + T^{6} \)
19 \( 1 - 18 T + 162 T^{2} - 883 T^{3} + 162 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - 9 T + 36 T^{2} - 18 p T^{3} + 207 T^{4} - 567 T^{5} + 1441 T^{6} - 567 p T^{7} + 207 p^{2} T^{8} - 18 p^{4} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T - 12 T^{2} - 19 T^{3} + 171 T^{4} + 18 p T^{5} - 1161 T^{6} + 18 p^{2} T^{7} + 171 p^{2} T^{8} - 19 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 30 T^{2} + 2 T^{3} + 570 T^{4} - 30 T^{5} - 7237 T^{6} - 30 p T^{7} + 570 p^{2} T^{8} + 2 p^{3} T^{9} - 30 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 3 T + 18 T^{2} + 102 T^{3} + 477 T^{4} + 1533 T^{5} + 6821 T^{6} + 1533 p T^{7} + 477 p^{2} T^{8} + 102 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T + 48 T^{2} + 142 T^{3} + 1407 T^{4} + 225 p T^{5} + 1597 p T^{6} + 225 p^{2} T^{7} + 1407 p^{2} T^{8} + 142 p^{3} T^{9} + 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 21 T + 210 T^{2} - 1258 T^{3} + 4431 T^{4} - 6129 T^{5} - 7357 T^{6} - 6129 p T^{7} + 4431 p^{2} T^{8} - 1258 p^{3} T^{9} + 210 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 18 T^{2} + 90 T^{3} + 81 T^{4} + 4413 T^{5} + 19153 T^{6} + 4413 p T^{7} + 81 p^{2} T^{8} + 90 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T - 12 T^{2} - 225 T^{3} + 987 T^{4} + 198 p T^{5} - 4417 T^{6} + 198 p^{2} T^{7} + 987 p^{2} T^{8} - 225 p^{3} T^{9} - 12 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 9 T + 54 T^{2} + 305 T^{3} + 54 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 177 T^{2} + 1657 T^{3} + 15216 T^{4} + 110376 T^{5} + 759557 T^{6} + 110376 p T^{7} + 15216 p^{2} T^{8} + 1657 p^{3} T^{9} + 177 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 99 T^{2} + 561 T^{3} + 8010 T^{4} + 34446 T^{5} + 422585 T^{6} + 34446 p T^{7} + 8010 p^{2} T^{8} + 561 p^{3} T^{9} + 99 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T + 63 T^{2} - 211 T^{3} + 684 T^{4} + 19926 T^{5} - 157555 T^{6} + 19926 p T^{7} + 684 p^{2} T^{8} - 211 p^{3} T^{9} + 63 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 12 T + 174 T^{2} - 1510 T^{3} + 13782 T^{4} - 123426 T^{5} + 886811 T^{6} - 123426 p T^{7} + 13782 p^{2} T^{8} - 1510 p^{3} T^{9} + 174 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 27 T + 324 T^{2} + 1940 T^{3} - 1485 T^{4} - 146097 T^{5} - 1529227 T^{6} - 146097 p T^{7} - 1485 p^{2} T^{8} + 1940 p^{3} T^{9} + 324 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 60 T^{2} + 806 T^{3} - 720 T^{4} - 27153 T^{5} + 261519 T^{6} - 27153 p T^{7} - 720 p^{2} T^{8} + 806 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 21 T + 126 T^{2} - 780 T^{3} - 13725 T^{4} - 32307 T^{5} + 306467 T^{6} - 32307 p T^{7} - 13725 p^{2} T^{8} - 780 p^{3} T^{9} + 126 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 39 T + 561 T^{2} + 2527 T^{3} - 21102 T^{4} - 349974 T^{5} - 2934895 T^{6} - 349974 p T^{7} - 21102 p^{2} T^{8} + 2527 p^{3} T^{9} + 561 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 36 T + 558 T^{2} - 4367 T^{3} + 1161 T^{4} + 426735 T^{5} - 5460783 T^{6} + 426735 p T^{7} + 1161 p^{2} T^{8} - 4367 p^{3} T^{9} + 558 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 45 T + 1125 T^{2} - 20185 T^{3} + 284850 T^{4} - 3300750 T^{5} + 32006361 T^{6} - 3300750 p T^{7} + 284850 p^{2} T^{8} - 20185 p^{3} T^{9} + 1125 p^{4} T^{10} - 45 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 27 T + 258 T^{2} - 2747 T^{3} + 48153 T^{4} - 422916 T^{5} + 2558915 T^{6} - 422916 p T^{7} + 48153 p^{2} T^{8} - 2747 p^{3} T^{9} + 258 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 30 T + 246 T^{2} - 1397 T^{3} - 27591 T^{4} + 122913 T^{5} + 4060949 T^{6} + 122913 p T^{7} - 27591 p^{2} T^{8} - 1397 p^{3} T^{9} + 246 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T + 3 T^{2} - 1129 T^{3} - 4671 T^{4} - 9261 T^{5} + 257046 T^{6} - 9261 p T^{7} - 4671 p^{2} T^{8} - 1129 p^{3} T^{9} + 3 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.24281585825146109694561140762, −5.17185111262734469171034311758, −5.15243122288700441071338657322, −5.04442535908172923999168695749, −4.85763839093879582831536877961, −4.77066081655070964270984549168, −4.76076498848460182381162928572, −4.05952900078461909307148297577, −3.97373687612345124653041119212, −3.54527994739407467823886440158, −3.51761039911733996323661124083, −3.45078067921121123017215990441, −3.28619461679563280672285440963, −3.00106478104452798650787986273, −2.80011279300455291717957663008, −2.58863457939569923675078311972, −2.53965684297629802787340502179, −2.46462490348901699069921742642, −1.95406402843603120860328260797, −1.67812860904528998213661484819, −1.65877458267209226100391789598, −1.42477986384020180366577454544, −1.10194035797027313256346712363, −0.945202638916765697906142328138, −0.43176456009906107629938940263, 0.43176456009906107629938940263, 0.945202638916765697906142328138, 1.10194035797027313256346712363, 1.42477986384020180366577454544, 1.65877458267209226100391789598, 1.67812860904528998213661484819, 1.95406402843603120860328260797, 2.46462490348901699069921742642, 2.53965684297629802787340502179, 2.58863457939569923675078311972, 2.80011279300455291717957663008, 3.00106478104452798650787986273, 3.28619461679563280672285440963, 3.45078067921121123017215990441, 3.51761039911733996323661124083, 3.54527994739407467823886440158, 3.97373687612345124653041119212, 4.05952900078461909307148297577, 4.76076498848460182381162928572, 4.77066081655070964270984549168, 4.85763839093879582831536877961, 5.04442535908172923999168695749, 5.15243122288700441071338657322, 5.17185111262734469171034311758, 5.24281585825146109694561140762

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

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