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Properties

Label 12-770e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.084\times 10^{17}$
Sign $1$
Analytic cond. $54026.5$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·4-s − 3·5-s − 2·8-s + 6·9-s − 9·10-s + 3·11-s + 12·13-s − 9·16-s − 12·17-s + 18·18-s + 6·19-s − 9·20-s + 9·22-s − 6·23-s + 3·25-s + 36·26-s + 2·27-s − 12·29-s − 9·32-s − 36·34-s + 18·36-s − 18·37-s + 18·38-s + 6·40-s + 48·41-s + 12·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 1.34·5-s − 0.707·8-s + 2·9-s − 2.84·10-s + 0.904·11-s + 3.32·13-s − 9/4·16-s − 2.91·17-s + 4.24·18-s + 1.37·19-s − 2.01·20-s + 1.91·22-s − 1.25·23-s + 3/5·25-s + 7.06·26-s + 0.384·27-s − 2.22·29-s − 1.59·32-s − 6.17·34-s + 3·36-s − 2.95·37-s + 2.91·38-s + 0.948·40-s + 7.49·41-s + 1.82·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(54026.5\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.80099284\)
\(L(\frac12)\) \(\approx\) \(10.80099284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 + 37 T^{3} + p^{3} T^{6} \)
11 \( ( 1 - T + T^{2} )^{3} \)
good3 \( 1 - 2 p T^{2} - 2 T^{3} + 2 p^{2} T^{4} + 2 p T^{5} - 53 T^{6} + 2 p^{2} T^{7} + 2 p^{4} T^{8} - 2 p^{3} T^{9} - 2 p^{5} T^{10} + p^{6} T^{12} \)
13 \( ( 1 - 6 T + 15 T^{2} - 20 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 12 T + 48 T^{2} + 234 T^{3} + 2076 T^{4} + 492 p T^{5} + 1307 p T^{6} + 492 p^{2} T^{7} + 2076 p^{2} T^{8} + 234 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 6 T + 24 T^{2} - 154 T^{3} + 162 T^{4} + 1062 T^{5} - 333 T^{6} + 1062 p T^{7} + 162 p^{2} T^{8} - 154 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 39 T^{2} + 342 T^{3} + 1158 T^{4} + 4542 T^{5} + 37483 T^{6} + 4542 p T^{7} + 1158 p^{2} T^{8} + 342 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 6 T + 24 T^{2} + 81 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 36 T^{2} - 214 T^{3} + 180 T^{4} + 3852 T^{5} + 30855 T^{6} + 3852 p T^{7} + 180 p^{2} T^{8} - 214 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 18 T + 126 T^{2} + 686 T^{3} + 5976 T^{4} + 45828 T^{5} + 277923 T^{6} + 45828 p T^{7} + 5976 p^{2} T^{8} + 686 p^{3} T^{9} + 126 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 24 T + 303 T^{2} - 2376 T^{3} + 303 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 6 T + 33 T^{2} - 92 T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 12 T + 111 T^{2} - 684 T^{3} + 1266 T^{4} + 15468 T^{5} - 140897 T^{6} + 15468 p T^{7} + 1266 p^{2} T^{8} - 684 p^{3} T^{9} + 111 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 150 T^{2} + 18 T^{3} + 14550 T^{4} - 1350 T^{5} - 894665 T^{6} - 1350 p T^{7} + 14550 p^{2} T^{8} + 18 p^{3} T^{9} - 150 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 12 T - 33 T^{2} - 324 T^{3} + 6198 T^{4} + 132 p T^{5} - 440741 T^{6} + 132 p^{2} T^{7} + 6198 p^{2} T^{8} - 324 p^{3} T^{9} - 33 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 102 T^{2} + 98 T^{3} + 7848 T^{4} + 14796 T^{5} - 627309 T^{6} + 14796 p T^{7} + 7848 p^{2} T^{8} + 98 p^{3} T^{9} - 102 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 15 T - 15 T^{2} - 166 T^{3} + 13275 T^{4} + 36855 T^{5} - 722010 T^{6} + 36855 p T^{7} + 13275 p^{2} T^{8} - 166 p^{3} T^{9} - 15 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 204 T^{2} - 9 T^{3} + 204 p T^{4} + p^{3} T^{6} )^{2} \)
73 \( 1 + 3 T - 165 T^{2} - 388 T^{3} + 16209 T^{4} + 18945 T^{5} - 1273050 T^{6} + 18945 p T^{7} + 16209 p^{2} T^{8} - 388 p^{3} T^{9} - 165 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T - 129 T^{2} - 532 T^{3} + 30114 T^{4} + 77076 T^{5} - 2175201 T^{6} + 77076 p T^{7} + 30114 p^{2} T^{8} - 532 p^{3} T^{9} - 129 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 24 T + 393 T^{2} + 4176 T^{3} + 393 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 12 T - 24 T^{2} + 414 T^{3} - 1308 T^{4} + 59676 T^{5} - 766397 T^{6} + 59676 p T^{7} - 1308 p^{2} T^{8} + 414 p^{3} T^{9} - 24 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 18 T + 387 T^{2} - 3644 T^{3} + 387 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.51337949273408683510098146722, −5.49313962969131029322023422857, −5.15610908763346434734229158424, −4.70250763735579129072389412007, −4.55689329348234688224564526805, −4.55398916188080026491709125293, −4.40967294547366977533950982696, −4.24650830881553165874716667843, −4.14523446000455361843267053214, −3.93354500556420829386624152723, −3.93276998936652661723860553616, −3.83974323224865568147710855787, −3.60124487963091757082956220586, −3.44544647469250009323419495424, −3.05121824404718893532232603686, −3.00779831493272973660905101418, −2.64954393651518450803603802840, −2.53362466259531859117986177573, −2.22783160967857542744316207399, −1.80689015865439450776989596138, −1.69076683723959721842828482822, −1.51407498424481858843173342193, −1.04275670089890098378935675216, −0.823740010754640125040292392454, −0.43535919690072113432607633239, 0.43535919690072113432607633239, 0.823740010754640125040292392454, 1.04275670089890098378935675216, 1.51407498424481858843173342193, 1.69076683723959721842828482822, 1.80689015865439450776989596138, 2.22783160967857542744316207399, 2.53362466259531859117986177573, 2.64954393651518450803603802840, 3.00779831493272973660905101418, 3.05121824404718893532232603686, 3.44544647469250009323419495424, 3.60124487963091757082956220586, 3.83974323224865568147710855787, 3.93276998936652661723860553616, 3.93354500556420829386624152723, 4.14523446000455361843267053214, 4.24650830881553165874716667843, 4.40967294547366977533950982696, 4.55398916188080026491709125293, 4.55689329348234688224564526805, 4.70250763735579129072389412007, 5.15610908763346434734229158424, 5.49313962969131029322023422857, 5.51337949273408683510098146722

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

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