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Properties

Label 12-637e6-1.1-c1e6-0-5
Degree $12$
Conductor $6.681\times 10^{16}$
Sign $1$
Analytic cond. $17318.0$
Root an. cond. $2.25532$
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  + 2·2-s + 4·3-s + 2·4-s + 5·5-s + 8·6-s + 7·9-s + 10·10-s + 4·11-s + 8·12-s + 6·13-s + 20·15-s + 4·17-s + 14·18-s + 7·19-s + 10·20-s + 8·22-s − 23-s + 18·25-s + 12·26-s − 14·29-s + 40·30-s − 3·31-s + 4·32-s + 16·33-s + 8·34-s + 14·36-s + 10·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 2.23·5-s + 3.26·6-s + 7/3·9-s + 3.16·10-s + 1.20·11-s + 2.30·12-s + 1.66·13-s + 5.16·15-s + 0.970·17-s + 3.29·18-s + 1.60·19-s + 2.23·20-s + 1.70·22-s − 0.208·23-s + 18/5·25-s + 2.35·26-s − 2.59·29-s + 7.30·30-s − 0.538·31-s + 0.707·32-s + 2.78·33-s + 1.37·34-s + 7/3·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(17318.0\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(61.35084930\)
\(L(\frac12)\) \(\approx\) \(61.35084930\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( ( 1 - T )^{6} \)
good2 \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{2} T^{5} - p^{2} T^{6} + p^{3} T^{7} - p^{4} T^{8} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 - 4 T + p^{2} T^{2} - 8 T^{3} - 4 p T^{4} + 62 T^{5} - 137 T^{6} + 62 p T^{7} - 4 p^{3} T^{8} - 8 p^{3} T^{9} + p^{6} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - p T + 7 T^{2} - 4 T^{3} + 19 T^{4} + T^{5} - 146 T^{6} + p T^{7} + 19 p^{2} T^{8} - 4 p^{3} T^{9} + 7 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T - 17 T^{2} + 40 T^{3} + 382 T^{4} - 38 p T^{5} - 3857 T^{6} - 38 p^{2} T^{7} + 382 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 33 T^{2} + 48 T^{3} + 1080 T^{4} - 470 T^{5} - 20731 T^{6} - 470 p T^{7} + 1080 p^{2} T^{8} + 48 p^{3} T^{9} - 33 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 7 T - 5 T^{2} + 28 T^{3} + 469 T^{4} + 875 T^{5} - 18166 T^{6} + 875 p T^{7} + 469 p^{2} T^{8} + 28 p^{3} T^{9} - 5 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + T - 27 T^{2} - 150 T^{3} + 51 T^{4} + 1733 T^{5} + 13694 T^{6} + 1733 p T^{7} + 51 p^{2} T^{8} - 150 p^{3} T^{9} - 27 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 3 T - 43 T^{2} - 118 T^{3} + 681 T^{4} + 335 T^{5} - 14122 T^{6} + 335 p T^{7} + 681 p^{2} T^{8} - 118 p^{3} T^{9} - 43 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 10 T - 19 T^{2} + 126 T^{3} + 3178 T^{4} - 3932 T^{5} - 126587 T^{6} - 3932 p T^{7} + 3178 p^{2} T^{8} + 126 p^{3} T^{9} - 19 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 6 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 17 T + 59 T^{2} - 420 T^{3} + 12425 T^{4} - 71363 T^{5} + 127970 T^{6} - 71363 p T^{7} + 12425 p^{2} T^{8} - 420 p^{3} T^{9} + 59 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 13 T - 29 T^{2} - 164 T^{3} + 10913 T^{4} + 34735 T^{5} - 380618 T^{6} + 34735 p T^{7} + 10913 p^{2} T^{8} - 164 p^{3} T^{9} - 29 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 22 T + 163 T^{2} - 1366 T^{3} + 21446 T^{4} - 157474 T^{5} + 757639 T^{6} - 157474 p T^{7} + 21446 p^{2} T^{8} - 1366 p^{3} T^{9} + 163 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 24 T + 233 T^{2} + 1928 T^{3} + 21078 T^{4} + 166136 T^{5} + 1069181 T^{6} + 166136 p T^{7} + 21078 p^{2} T^{8} + 1928 p^{3} T^{9} + 233 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 14 T + 31 T^{2} + 146 T^{3} - 1022 T^{4} + 44074 T^{5} - 594409 T^{6} + 44074 p T^{7} - 1022 p^{2} T^{8} + 146 p^{3} T^{9} + 31 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 4 T + 169 T^{2} - 374 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 5 T + 25 T^{2} - 1662 T^{3} + 4155 T^{4} - 5 p^{2} T^{5} + 1335370 T^{6} - 5 p^{3} T^{7} + 4155 p^{2} T^{8} - 1662 p^{3} T^{9} + 25 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + T - 167 T^{2} - 346 T^{3} + 14695 T^{4} + 22873 T^{5} - 1183810 T^{6} + 22873 p T^{7} + 14695 p^{2} T^{8} - 346 p^{3} T^{9} - 167 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 23 T + 376 T^{2} + 4021 T^{3} + 376 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 11 T - 91 T^{2} - 1626 T^{3} + 4307 T^{4} + 91583 T^{5} + 367922 T^{6} + 91583 p T^{7} + 4307 p^{2} T^{8} - 1626 p^{3} T^{9} - 91 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 3 T + 220 T^{2} - 575 T^{3} + 220 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.71582689338271742010158067868, −5.67122503739849240914972191275, −5.56761446432212740741097057825, −5.37215047726054079495927231925, −4.70998185669572303304709404730, −4.68944052630972722170433019945, −4.62470290957021935952555385931, −4.61416621282429080495741762374, −4.20936416378128946540106961963, −3.83257658587863719919889198070, −3.71405324026360066855495574597, −3.58972026829573545022849499640, −3.50262948687422162439013602052, −3.47822152447108134984713502529, −3.18823143946371564697327092321, −2.96550861823738999045185561556, −2.64590543205261566460566741738, −2.47256243152806849617862955026, −2.39155427785449701940472030599, −2.08332049483295383558027868615, −1.91307008976626801927001923979, −1.58387225089837139744746406110, −1.31978081998146039087572749674, −1.08210350213104928818412929754, −0.876548290787904101379704083952, 0.876548290787904101379704083952, 1.08210350213104928818412929754, 1.31978081998146039087572749674, 1.58387225089837139744746406110, 1.91307008976626801927001923979, 2.08332049483295383558027868615, 2.39155427785449701940472030599, 2.47256243152806849617862955026, 2.64590543205261566460566741738, 2.96550861823738999045185561556, 3.18823143946371564697327092321, 3.47822152447108134984713502529, 3.50262948687422162439013602052, 3.58972026829573545022849499640, 3.71405324026360066855495574597, 3.83257658587863719919889198070, 4.20936416378128946540106961963, 4.61416621282429080495741762374, 4.62470290957021935952555385931, 4.68944052630972722170433019945, 4.70998185669572303304709404730, 5.37215047726054079495927231925, 5.56761446432212740741097057825, 5.67122503739849240914972191275, 5.71582689338271742010158067868

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

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