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Properties

Label 12-693e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.108\times 10^{17}$
Sign $1$
Analytic cond. $28711.9$
Root an. cond. $2.35236$
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  + 4-s − 2·5-s + 2·7-s + 6·8-s − 3·11-s − 22·13-s + 2·16-s − 3·17-s + 11·19-s − 2·20-s + 12·23-s + 8·25-s + 2·28-s + 18·29-s + 3·31-s + 9·32-s − 4·35-s + 4·37-s − 12·40-s + 10·41-s + 4·43-s − 3·44-s − 3·47-s − 10·49-s − 22·52-s + 17·53-s + 6·55-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 0.755·7-s + 2.12·8-s − 0.904·11-s − 6.10·13-s + 1/2·16-s − 0.727·17-s + 2.52·19-s − 0.447·20-s + 2.50·23-s + 8/5·25-s + 0.377·28-s + 3.34·29-s + 0.538·31-s + 1.59·32-s − 0.676·35-s + 0.657·37-s − 1.89·40-s + 1.56·41-s + 0.609·43-s − 0.452·44-s − 0.437·47-s − 1.42·49-s − 3.05·52-s + 2.33·53-s + 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(28711.9\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.531605590\)
\(L(\frac12)\) \(\approx\) \(6.531605590\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 2 T + 2 p T^{2} - 23 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11 \( ( 1 + T + T^{2} )^{3} \)
good2 \( 1 - T^{2} - 3 p T^{3} - T^{4} + 3 T^{5} + 23 T^{6} + 3 p T^{7} - p^{2} T^{8} - 3 p^{4} T^{9} - p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 2 T - 4 T^{2} - 14 T^{3} - 6 T^{4} + 2 p T^{5} + 11 p T^{6} + 2 p^{2} T^{7} - 6 p^{2} T^{8} - 14 p^{3} T^{9} - 4 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 11 T + 75 T^{2} + 321 T^{3} + 75 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 3 T - 40 T^{2} - 43 T^{3} + 1283 T^{4} + 524 T^{5} - 24567 T^{6} + 524 p T^{7} + 1283 p^{2} T^{8} - 43 p^{3} T^{9} - 40 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 44 T^{2} - 125 T^{3} + 495 T^{4} + 22 p T^{5} - 647 p T^{6} + 22 p^{2} T^{7} + 495 p^{2} T^{8} - 125 p^{3} T^{9} + 44 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 32 T^{2} - 146 T^{3} + 2780 T^{4} - 11612 T^{5} + 18447 T^{6} - 11612 p T^{7} + 2780 p^{2} T^{8} - 146 p^{3} T^{9} + 32 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 9 T + 67 T^{2} - 469 T^{3} + 67 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 3 T - 40 T^{2} + 11 T^{3} + 645 T^{4} + 2360 T^{5} - 17257 T^{6} + 2360 p T^{7} + 645 p^{2} T^{8} + 11 p^{3} T^{9} - 40 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T - 59 T^{2} - 12 T^{3} + 2274 T^{4} + 5924 T^{5} - 107987 T^{6} + 5924 p T^{7} + 2274 p^{2} T^{8} - 12 p^{3} T^{9} - 59 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 5 T + 43 T^{2} - 301 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 2 T + 104 T^{2} - 131 T^{3} + 104 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 3 T - 130 T^{2} - 133 T^{3} + 11963 T^{4} + 5654 T^{5} - 644697 T^{6} + 5654 p T^{7} + 11963 p^{2} T^{8} - 133 p^{3} T^{9} - 130 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 17 T + 56 T^{2} - 315 T^{3} + 10949 T^{4} - 52646 T^{5} - 68035 T^{6} - 52646 p T^{7} + 10949 p^{2} T^{8} - 315 p^{3} T^{9} + 56 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 8 T + 44 T^{2} - 918 T^{3} + 2252 T^{4} - 1388 T^{5} + 308015 T^{6} - 1388 p T^{7} + 2252 p^{2} T^{8} - 918 p^{3} T^{9} + 44 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 24 T + 221 T^{2} - 1912 T^{3} + 25074 T^{4} - 222784 T^{5} + 1539557 T^{6} - 222784 p T^{7} + 25074 p^{2} T^{8} - 1912 p^{3} T^{9} + 221 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T - 13 T^{2} + 128 T^{3} + 18882 T^{4} - 91192 T^{5} - 485189 T^{6} - 91192 p T^{7} + 18882 p^{2} T^{8} + 128 p^{3} T^{9} - 13 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 127 T^{2} + 575 T^{3} + 127 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 20 T + 156 T^{2} - 290 T^{3} - 4176 T^{4} + 45920 T^{5} - 432669 T^{6} + 45920 p T^{7} - 4176 p^{2} T^{8} - 290 p^{3} T^{9} + 156 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 3 T - 190 T^{2} - 69 T^{3} + 22881 T^{4} - 9336 T^{5} - 2087681 T^{6} - 9336 p T^{7} + 22881 p^{2} T^{8} - 69 p^{3} T^{9} - 190 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 11 T + 265 T^{2} - 1823 T^{3} + 265 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - T - 258 T^{2} + 91 T^{3} + 43855 T^{4} - 7144 T^{5} - 4537567 T^{6} - 7144 p T^{7} + 43855 p^{2} T^{8} + 91 p^{3} T^{9} - 258 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 279 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.35507625661096826105101103357, −5.29487797921351940483815216389, −5.12023971712807297228964345865, −5.08380877763248858249569337224, −4.82204155482329666228820784666, −4.79770324323920626913904437356, −4.78902684116010550395826980833, −4.51022384425030009955867756896, −4.39030731057562813620400017103, −4.10689318302230619976533362654, −4.04800180638069445903359698884, −3.46187377898795080402581715970, −3.37398482194765904418597686468, −3.29116946092428942354737408537, −2.86179139187953556390210927162, −2.61399679445366661060172809454, −2.57361372643907765114430891355, −2.54479652341114986569214699365, −2.36094971793377371762365416199, −2.05821715277273023202709714564, −1.87477741180211773820985903637, −1.15493543521656163124134729917, −0.974599347872131213494775033514, −0.71168542978710544177203465119, −0.69277665132456341794769097435, 0.69277665132456341794769097435, 0.71168542978710544177203465119, 0.974599347872131213494775033514, 1.15493543521656163124134729917, 1.87477741180211773820985903637, 2.05821715277273023202709714564, 2.36094971793377371762365416199, 2.54479652341114986569214699365, 2.57361372643907765114430891355, 2.61399679445366661060172809454, 2.86179139187953556390210927162, 3.29116946092428942354737408537, 3.37398482194765904418597686468, 3.46187377898795080402581715970, 4.04800180638069445903359698884, 4.10689318302230619976533362654, 4.39030731057562813620400017103, 4.51022384425030009955867756896, 4.78902684116010550395826980833, 4.79770324323920626913904437356, 4.82204155482329666228820784666, 5.08380877763248858249569337224, 5.12023971712807297228964345865, 5.29487797921351940483815216389, 5.35507625661096826105101103357

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

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