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Properties

Label 12-114e6-1.1-c1e6-0-3
Degree $12$
Conductor $2.195\times 10^{12}$
Sign $1$
Analytic cond. $0.568973$
Root an. cond. $0.954093$
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 + 8-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 + 9·40-s − 15·41-s + 3·43-s − 9·47-s + 21·49-s + 12·53-s + 3·56-s + 27·59-s + 3·61-s − 27·65-s + 21·67-s + 39·71-s + 36·73-s − 45·79-s + ⋯
L(s)  = 1  + 4.02·5-s + 1.13·7-s + 0.353·8-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 + 1.42·40-s − 2.34·41-s + 0.457·43-s − 1.31·47-s + 3·49-s + 1.64·53-s + 0.400·56-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 + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.568973\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.069027013\)
\(L(\frac12)\) \(\approx\) \(2.069027013\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
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

−7.998941167135848100276310960522, −7.19677649520715541997633418116, −7.02887408707105239057327372450, −6.81491205309252606353575066094, −6.64844246915801982456791336779, −6.64187444688808074501151718505, −6.63272809703913071492021040875, −6.25259008286742400940574054806, −5.74595489274388053617687269860, −5.68367552208875440906799089145, −5.47053757772606299681077394104, −5.41114720546800750565035612468, −5.27375097732845350294000050430, −5.02212734313338549169827739806, −4.36028722621144727898319685301, −4.27285914981551215958887180856, −4.19197745442549323355301099569, −3.92441416291297868357236060348, −3.56868407894679130114210868349, −2.62755000384875629765436946495, −2.47989876183043574185938126294, −2.27016040909711471604506682140, −1.98266696372122383748274164302, −1.93162657894447532058412909292, −1.64200822936618385186842858496, 1.64200822936618385186842858496, 1.93162657894447532058412909292, 1.98266696372122383748274164302, 2.27016040909711471604506682140, 2.47989876183043574185938126294, 2.62755000384875629765436946495, 3.56868407894679130114210868349, 3.92441416291297868357236060348, 4.19197745442549323355301099569, 4.27285914981551215958887180856, 4.36028722621144727898319685301, 5.02212734313338549169827739806, 5.27375097732845350294000050430, 5.41114720546800750565035612468, 5.47053757772606299681077394104, 5.68367552208875440906799089145, 5.74595489274388053617687269860, 6.25259008286742400940574054806, 6.63272809703913071492021040875, 6.64187444688808074501151718505, 6.64844246915801982456791336779, 6.81491205309252606353575066094, 7.02887408707105239057327372450, 7.19677649520715541997633418116, 7.998941167135848100276310960522

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

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