Properties

Label 12-114e6-1.1-c1e6-0-0
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  − 3·5-s + 3·7-s − 8-s + 12·11-s − 21·13-s + 3·17-s + 6·19-s − 15·23-s + 9·25-s − 27-s + 15·29-s + 3·31-s − 9·35-s + 6·37-s + 3·40-s − 9·41-s − 9·43-s − 21·47-s + 15·49-s + 30·53-s − 36·55-s − 3·56-s + 27·59-s − 9·61-s + 63·65-s − 15·67-s + 9·71-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.13·7-s − 0.353·8-s + 3.61·11-s − 5.82·13-s + 0.727·17-s + 1.37·19-s − 3.12·23-s + 9/5·25-s − 0.192·27-s + 2.78·29-s + 0.538·31-s − 1.52·35-s + 0.986·37-s + 0.474·40-s − 1.40·41-s − 1.37·43-s − 3.06·47-s + 15/7·49-s + 4.12·53-s − 4.85·55-s − 0.400·56-s + 3.51·59-s − 1.15·61-s + 7.81·65-s − 1.83·67-s + 1.06·71-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: induced by $\chi_{114} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(0.8404722226\)
\(L(\frac12)\) \(\approx\) \(0.8404722226\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 3 T - 9 T^{4} - 3 p T^{5} + 109 T^{6} - 3 p^{2} T^{7} - 9 p^{2} T^{8} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T - 6 T^{2} + 41 T^{3} - 9 T^{4} - 162 T^{5} + 519 T^{6} - 162 p T^{7} - 9 p^{2} T^{8} + 41 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 12 T + 6 p T^{2} - 306 T^{3} + 1446 T^{4} - 5430 T^{5} + 17539 T^{6} - 5430 p T^{7} + 1446 p^{2} T^{8} - 306 p^{3} T^{9} + 6 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 7 T + p T^{2} )^{3}( 1 - 89 T^{3} + p^{3} T^{6} ) \)
17 \( 1 - 3 T - 18 T^{2} + 180 T^{3} - 423 T^{4} - 105 p T^{5} + 1097 p T^{6} - 105 p^{2} T^{7} - 423 p^{2} T^{8} + 180 p^{3} T^{9} - 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 108 T^{2} + 522 T^{3} + 1881 T^{4} + 6855 T^{5} + 32347 T^{6} + 6855 p T^{7} + 1881 p^{2} T^{8} + 522 p^{3} T^{9} + 108 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 15 T + 72 T^{2} + 36 T^{3} - 729 T^{4} - 13389 T^{5} + 142381 T^{6} - 13389 p T^{7} - 729 p^{2} T^{8} + 36 p^{3} T^{9} + 72 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 78 T^{2} + 77 T^{3} + 4365 T^{4} - 1404 T^{5} - 155145 T^{6} - 1404 p T^{7} + 4365 p^{2} T^{8} + 77 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 3 T + 102 T^{2} - 203 T^{3} + 102 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 9 T + 27 T^{2} + 153 T^{3} - 1728 T^{4} - 21024 T^{5} - 78155 T^{6} - 21024 p T^{7} - 1728 p^{2} T^{8} + 153 p^{3} T^{9} + 27 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T + 45 T^{2} + 209 T^{3} - 972 T^{4} - 25110 T^{5} - 174579 T^{6} - 25110 p T^{7} - 972 p^{2} T^{8} + 209 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 21 T + 261 T^{2} + 2673 T^{3} + 22680 T^{4} + 170022 T^{5} + 1202725 T^{6} + 170022 p T^{7} + 22680 p^{2} T^{8} + 2673 p^{3} T^{9} + 261 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 30 T + 378 T^{2} - 2214 T^{3} - 3276 T^{4} + 201552 T^{5} - 2070449 T^{6} + 201552 p T^{7} - 3276 p^{2} T^{8} - 2214 p^{3} T^{9} + 378 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 27 T + 360 T^{2} - 3312 T^{3} + 30879 T^{4} - 316287 T^{5} + 2798533 T^{6} - 316287 p T^{7} + 30879 p^{2} T^{8} - 3312 p^{3} T^{9} + 360 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T + 72 T^{2} + 506 T^{3} + 108 p T^{4} + 59805 T^{5} + 418827 T^{6} + 59805 p T^{7} + 108 p^{3} T^{8} + 506 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 15 T + 156 T^{2} + 1472 T^{3} + 16461 T^{4} + 143415 T^{5} + 1174659 T^{6} + 143415 p T^{7} + 16461 p^{2} T^{8} + 1472 p^{3} T^{9} + 156 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 9 T + 81 T^{2} - 9 T^{3} + 2916 T^{4} + 32418 T^{5} - 92555 T^{6} + 32418 p T^{7} + 2916 p^{2} T^{8} - 9 p^{3} T^{9} + 81 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 12 T + 246 T^{2} - 2623 T^{3} + 30177 T^{4} - 326385 T^{5} + 2687649 T^{6} - 326385 p T^{7} + 30177 p^{2} T^{8} - 2623 p^{3} T^{9} + 246 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 15 T + 105 T^{2} - 559 T^{3} - 6300 T^{4} + 127350 T^{5} - 1087083 T^{6} + 127350 p T^{7} - 6300 p^{2} T^{8} - 559 p^{3} T^{9} + 105 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 3 T + 30 T^{2} + 2727 T^{3} + 6369 T^{4} + 60420 T^{5} + 2990995 T^{6} + 60420 p T^{7} + 6369 p^{2} T^{8} + 2727 p^{3} T^{9} + 30 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 48 T + 1044 T^{2} + 11907 T^{3} + 43551 T^{4} - 760371 T^{5} - 12833135 T^{6} - 760371 p T^{7} + 43551 p^{2} T^{8} + 11907 p^{3} T^{9} + 1044 p^{4} T^{10} + 48 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 99 T^{2} + 551 T^{3} - 5967 T^{4} - 122661 T^{5} + 2662518 T^{6} - 122661 p T^{7} - 5967 p^{2} T^{8} + 551 p^{3} T^{9} + 99 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.60372561750819320118854880403, −7.53738899281341391381666165037, −7.09585023304894204786799674667, −7.04490596590983391683773288776, −6.91554202314556867885778213394, −6.80229014550440123008964503563, −6.67951729380984350864459846999, −6.25724682726125771144032170848, −5.98635482940148055999115258791, −5.63968843541801313064519084092, −5.36251668658551649277159267406, −5.15577027918857297465035590924, −4.88378225940697214772082102543, −4.87480381338100059325188897645, −4.50831837217890108253404145998, −4.24712382518533956421844402942, −4.09594156929403499503905162674, −3.81164061229600247761355784863, −3.64104005372801854691714920830, −2.95248436050377181495301089302, −2.87768604156883120626606761591, −2.47686776152535551816621986439, −2.11885032600494272846516133560, −1.67591113988167962295001015136, −0.861446573471696353256315581864, 0.861446573471696353256315581864, 1.67591113988167962295001015136, 2.11885032600494272846516133560, 2.47686776152535551816621986439, 2.87768604156883120626606761591, 2.95248436050377181495301089302, 3.64104005372801854691714920830, 3.81164061229600247761355784863, 4.09594156929403499503905162674, 4.24712382518533956421844402942, 4.50831837217890108253404145998, 4.87480381338100059325188897645, 4.88378225940697214772082102543, 5.15577027918857297465035590924, 5.36251668658551649277159267406, 5.63968843541801313064519084092, 5.98635482940148055999115258791, 6.25724682726125771144032170848, 6.67951729380984350864459846999, 6.80229014550440123008964503563, 6.91554202314556867885778213394, 7.04490596590983391683773288776, 7.09585023304894204786799674667, 7.53738899281341391381666165037, 7.60372561750819320118854880403

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

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