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Properties

Label 12-114e6-1.1-c1e6-0-2
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 + 9·13-s − 3·17-s + 27·23-s − 3·25-s + 27-s − 3·29-s − 15·31-s − 9·35-s − 6·37-s − 3·40-s − 15·41-s + 3·43-s + 15·47-s + 3·49-s + 6·53-s + 3·56-s − 27·59-s − 15·61-s + 27·65-s − 3·67-s + 3·71-s + 12·73-s + 27·79-s + 3·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.13·7-s − 0.353·8-s + 2.49·13-s − 0.727·17-s + 5.62·23-s − 3/5·25-s + 0.192·27-s − 0.557·29-s − 2.69·31-s − 1.52·35-s − 0.986·37-s − 0.474·40-s − 2.34·41-s + 0.457·43-s + 2.18·47-s + 3/7·49-s + 0.824·53-s + 0.400·56-s − 3.51·59-s − 1.92·61-s + 3.34·65-s − 0.366·67-s + 0.356·71-s + 1.40·73-s + 3.03·79-s + 0.329·83-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

\(L(1)\) \(\approx\) \(1.312955074\)
\(L(\frac12)\) \(\approx\) \(1.312955074\)
\(L(\frac{3}{2})\) 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 \( 1 + 107 T^{3} + p^{3} T^{6} \)
good5 \( 1 - 3 T + 12 T^{2} - 16 T^{3} + 3 p^{2} T^{4} - 153 T^{5} + 581 T^{6} - 153 p T^{7} + 3 p^{4} T^{8} - 16 p^{3} T^{9} + 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T + 6 T^{2} + 39 T^{3} + 33 T^{4} - 24 T^{5} + 407 T^{6} - 24 p T^{7} + 33 p^{2} T^{8} + 39 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 12 T^{2} + 74 T^{3} + 12 T^{4} - 444 T^{5} + 2447 T^{6} - 444 p T^{7} + 12 p^{2} T^{8} + 74 p^{3} T^{9} - 12 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 106 T^{3} + 27 T^{4} + 1161 T^{5} - 4731 T^{6} + 1161 p T^{7} + 27 p^{2} T^{8} - 106 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T + 18 T^{2} + 126 T^{3} + 621 T^{4} + 2217 T^{5} + 10549 T^{6} + 2217 p T^{7} + 621 p^{2} T^{8} + 126 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 27 T + 378 T^{2} - 3694 T^{3} + 28215 T^{4} - 177039 T^{5} + 927971 T^{6} - 177039 p T^{7} + 28215 p^{2} T^{8} - 3694 p^{3} T^{9} + 378 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 12 T^{2} - 128 T^{3} + 147 T^{4} - 3231 T^{5} - 1507 T^{6} - 3231 p T^{7} + 147 p^{2} T^{8} - 128 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 15 T + 60 T^{2} + 397 T^{3} + 6525 T^{4} + 32400 T^{5} + 78903 T^{6} + 32400 p T^{7} + 6525 p^{2} T^{8} + 397 p^{3} T^{9} + 60 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 66 T^{2} + 239 T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 87 T^{2} - 53 T^{3} - 2874 T^{4} - 20052 T^{5} - 101167 T^{6} - 20052 p T^{7} - 2874 p^{2} T^{8} - 53 p^{3} T^{9} + 87 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 3 T^{2} - 107 T^{3} - 1206 T^{4} + 3258 T^{5} + 55089 T^{6} + 3258 p T^{7} - 1206 p^{2} T^{8} - 107 p^{3} T^{9} - 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 51 T^{2} + 503 T^{3} - 4224 T^{4} - 24030 T^{5} + 424469 T^{6} - 24030 p T^{7} - 4224 p^{2} T^{8} + 503 p^{3} T^{9} + 51 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 30 T^{2} - 226 T^{3} + 2688 T^{4} - 11340 T^{5} + 185231 T^{6} - 11340 p T^{7} + 2688 p^{2} T^{8} - 226 p^{3} T^{9} - 30 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 27 T + 198 T^{2} - 1134 T^{3} - 21663 T^{4} - 27117 T^{5} + 768025 T^{6} - 27117 p T^{7} - 21663 p^{2} T^{8} - 1134 p^{3} T^{9} + 198 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 15 T + 84 T^{2} + 32 T^{3} - 1638 T^{4} - 9657 T^{5} - 118173 T^{6} - 9657 p T^{7} - 1638 p^{2} T^{8} + 32 p^{3} T^{9} + 84 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T + 72 T^{2} + 30 T^{3} + 4257 T^{4} - 9465 T^{5} + 51479 T^{6} - 9465 p T^{7} + 4257 p^{2} T^{8} + 30 p^{3} T^{9} + 72 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T - 87 T^{2} - 187 T^{3} - 2094 T^{4} + 26118 T^{5} + 499145 T^{6} + 26118 p T^{7} - 2094 p^{2} T^{8} - 187 p^{3} T^{9} - 87 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 12 T + 54 T^{2} - 795 T^{3} - 135 T^{4} + 40083 T^{5} + 66905 T^{6} + 40083 p T^{7} - 135 p^{2} T^{8} - 795 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 27 T + 351 T^{2} - 3337 T^{3} + 19872 T^{4} - 49734 T^{5} + 58989 T^{6} - 49734 p T^{7} + 19872 p^{2} T^{8} - 3337 p^{3} T^{9} + 351 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 3 T - 204 T^{2} + 459 T^{3} + 26043 T^{4} - 31530 T^{5} - 2382653 T^{6} - 31530 p T^{7} + 26043 p^{2} T^{8} + 459 p^{3} T^{9} - 204 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 42 T + 756 T^{2} - 7371 T^{3} + 29799 T^{4} + 258951 T^{5} - 4921811 T^{6} + 258951 p T^{7} + 29799 p^{2} T^{8} - 7371 p^{3} T^{9} + 756 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 171 T^{2} - 713 T^{3} - 4239 T^{4} + 76491 T^{5} - 635802 T^{6} + 76491 p T^{7} - 4239 p^{2} T^{8} - 713 p^{3} T^{9} + 171 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.72747321777651010138643965543, −7.41832664270459930427835493549, −7.21277773121902388004798520111, −6.92131650133965409879110128299, −6.77539340786880919906913353075, −6.55574575181375588999787034321, −6.37797401649676155501937940650, −6.27331503240779594881168596759, −6.05060700637369234355671370711, −5.81581155854740865283464263021, −5.43732285535737125663664636832, −5.25724144962815790430590636902, −5.03566724736339800212697429513, −5.02949662776324935903552890573, −4.76631647826269197688479663170, −4.06590607101284889854178893659, −3.86957946349673115241725355650, −3.56332128718559954522635432915, −3.51291803244063787848342247146, −3.14228509481876831132742873618, −2.96397647136189785474413584675, −2.42425829885683215657809595872, −2.13048373333180802412150150202, −1.42302436248626201470600860177, −1.28242763175821312831686930381, 1.28242763175821312831686930381, 1.42302436248626201470600860177, 2.13048373333180802412150150202, 2.42425829885683215657809595872, 2.96397647136189785474413584675, 3.14228509481876831132742873618, 3.51291803244063787848342247146, 3.56332128718559954522635432915, 3.86957946349673115241725355650, 4.06590607101284889854178893659, 4.76631647826269197688479663170, 5.02949662776324935903552890573, 5.03566724736339800212697429513, 5.25724144962815790430590636902, 5.43732285535737125663664636832, 5.81581155854740865283464263021, 6.05060700637369234355671370711, 6.27331503240779594881168596759, 6.37797401649676155501937940650, 6.55574575181375588999787034321, 6.77539340786880919906913353075, 6.92131650133965409879110128299, 7.21277773121902388004798520111, 7.41832664270459930427835493549, 7.72747321777651010138643965543

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

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