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Properties

Label 12-287e6-1.1-c1e6-0-0
Degree $12$
Conductor $5.588\times 10^{14}$
Sign $1$
Analytic cond. $144.861$
Root an. cond. $1.51383$
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-s − 4·3-s − 4-s − 5-s + 4·6-s − 6·7-s + 3·8-s + 6·9-s + 10-s + 6·11-s + 4·12-s + 7·13-s + 6·14-s + 4·15-s − 4·16-s + 7·17-s − 6·18-s + 2·19-s + 20-s + 24·21-s − 6·22-s + 20·23-s − 12·24-s − 7·26-s + 2·27-s + 6·28-s − 9·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 1/2·4-s − 0.447·5-s + 1.63·6-s − 2.26·7-s + 1.06·8-s + 2·9-s + 0.316·10-s + 1.80·11-s + 1.15·12-s + 1.94·13-s + 1.60·14-s + 1.03·15-s − 16-s + 1.69·17-s − 1.41·18-s + 0.458·19-s + 0.223·20-s + 5.23·21-s − 1.27·22-s + 4.17·23-s − 2.44·24-s − 1.37·26-s + 0.384·27-s + 1.13·28-s − 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 41^{6}\)
Sign: $1$
Analytic conductor: \(144.861\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 41^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3106369844\)
\(L(\frac12)\) \(\approx\) \(0.3106369844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T )^{6} \)
41 \( ( 1 - T )^{6} \)
good2 \( 1 + T + p T^{2} + 3 T^{4} + p^{2} T^{5} + 17 T^{6} + p^{3} T^{7} + 3 p^{2} T^{8} + p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + 4 T + 10 T^{2} + 14 T^{3} + 26 T^{4} + 19 p T^{5} + 130 T^{6} + 19 p^{2} T^{7} + 26 p^{2} T^{8} + 14 p^{3} T^{9} + 10 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + T + T^{2} + 9 T^{3} - p T^{4} - 6 T^{5} + 134 T^{6} - 6 p T^{7} - p^{3} T^{8} + 9 p^{3} T^{9} + p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 37 T^{2} - 112 T^{3} + 567 T^{4} - 1994 T^{5} + 8902 T^{6} - 1994 p T^{7} + 567 p^{2} T^{8} - 112 p^{3} T^{9} + 37 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 7 T + 29 T^{2} - 125 T^{3} + 387 T^{4} - 877 T^{5} + 3108 T^{6} - 877 p T^{7} + 387 p^{2} T^{8} - 125 p^{3} T^{9} + 29 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 7 T + 105 T^{2} - 569 T^{3} + 267 p T^{4} - 18913 T^{5} + 103464 T^{6} - 18913 p T^{7} + 267 p^{3} T^{8} - 569 p^{3} T^{9} + 105 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 46 T^{2} - 42 T^{3} + 1172 T^{4} + 383 T^{5} + 21830 T^{6} + 383 p T^{7} + 1172 p^{2} T^{8} - 42 p^{3} T^{9} + 46 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 20 T + 290 T^{2} - 2858 T^{3} + 22970 T^{4} - 145271 T^{5} + 774478 T^{6} - 145271 p T^{7} + 22970 p^{2} T^{8} - 2858 p^{3} T^{9} + 290 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 95 T^{2} + 533 T^{3} + 3647 T^{4} + 18534 T^{5} + 110962 T^{6} + 18534 p T^{7} + 3647 p^{2} T^{8} + 533 p^{3} T^{9} + 95 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 27 T + 443 T^{2} + 5167 T^{3} + 47219 T^{4} + 349172 T^{5} + 2134434 T^{6} + 349172 p T^{7} + 47219 p^{2} T^{8} + 5167 p^{3} T^{9} + 443 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 19 T + 276 T^{2} - 3096 T^{3} + 28095 T^{4} - 214681 T^{5} + 1424272 T^{6} - 214681 p T^{7} + 28095 p^{2} T^{8} - 3096 p^{3} T^{9} + 276 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 19 T + 285 T^{2} - 3095 T^{3} + 29793 T^{4} - 236989 T^{5} + 39466 p T^{6} - 236989 p T^{7} + 29793 p^{2} T^{8} - 3095 p^{3} T^{9} + 285 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 19 T + 8 p T^{2} + 4402 T^{3} + 50057 T^{4} + 412039 T^{5} + 3251324 T^{6} + 412039 p T^{7} + 50057 p^{2} T^{8} + 4402 p^{3} T^{9} + 8 p^{5} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 5 T + 191 T^{2} - 861 T^{3} + 19247 T^{4} - 75162 T^{5} + 1236466 T^{6} - 75162 p T^{7} + 19247 p^{2} T^{8} - 861 p^{3} T^{9} + 191 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 7 T + 171 T^{2} + 589 T^{3} + 9235 T^{4} - 15790 T^{5} + 310242 T^{6} - 15790 p T^{7} + 9235 p^{2} T^{8} + 589 p^{3} T^{9} + 171 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 3 p T^{2} + 1050 T^{3} + 10135 T^{4} + 11674 T^{5} + 272594 T^{6} + 11674 p T^{7} + 10135 p^{2} T^{8} + 1050 p^{3} T^{9} + 3 p^{5} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 27 T + 645 T^{2} - 9607 T^{3} + 129139 T^{4} - 1305704 T^{5} + 12088494 T^{6} - 1305704 p T^{7} + 129139 p^{2} T^{8} - 9607 p^{3} T^{9} + 645 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 321 T^{2} + 1680 T^{3} + 48347 T^{4} + 214362 T^{5} + 4349446 T^{6} + 214362 p T^{7} + 48347 p^{2} T^{8} + 1680 p^{3} T^{9} + 321 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 52 T + 1447 T^{2} - 27846 T^{3} + 408023 T^{4} - 4752206 T^{5} + 44927922 T^{6} - 4752206 p T^{7} + 408023 p^{2} T^{8} - 27846 p^{3} T^{9} + 1447 p^{4} T^{10} - 52 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 322 T^{2} - 96 T^{3} + 49695 T^{4} - 14720 T^{5} + 4820732 T^{6} - 14720 p T^{7} + 49695 p^{2} T^{8} - 96 p^{3} T^{9} + 322 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 12 T + 371 T^{2} - 4410 T^{3} + 66551 T^{4} - 681302 T^{5} + 7059658 T^{6} - 681302 p T^{7} + 66551 p^{2} T^{8} - 4410 p^{3} T^{9} + 371 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 38 T + 938 T^{2} + 15970 T^{3} + 222014 T^{4} + 2537075 T^{5} + 25723552 T^{6} + 2537075 p T^{7} + 222014 p^{2} T^{8} + 15970 p^{3} T^{9} + 938 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 8 T + 266 T^{2} - 1362 T^{3} + 34864 T^{4} - 189883 T^{5} + 3886868 T^{6} - 189883 p T^{7} + 34864 p^{2} T^{8} - 1362 p^{3} T^{9} + 266 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.46661727348123593781891420818, −6.29906134716818663296225780485, −6.06361141705955454314960532953, −5.83337267646263694445845412637, −5.72514990553029491632028922693, −5.59759352229335320574216629958, −5.54542361702429454275172415652, −5.31807492551881003216459297810, −4.99084112716440111855460068245, −4.72943244324318017616117073121, −4.62378838901466027699821221400, −4.46345831168935850252696587765, −3.86153618672425635165382686946, −3.77309444083335479053955210645, −3.72523009581308654023372077230, −3.61187020248674200836956173386, −3.33620011226421082636492671267, −3.17653546942301702643588902945, −2.64553041625941102682902178844, −2.51833126683717104335404106352, −1.94750764742205736906413678243, −1.27871960126229417602407470457, −1.09360493080290017379862816227, −1.03985454354036588510666576829, −0.38268555086796134873457892850, 0.38268555086796134873457892850, 1.03985454354036588510666576829, 1.09360493080290017379862816227, 1.27871960126229417602407470457, 1.94750764742205736906413678243, 2.51833126683717104335404106352, 2.64553041625941102682902178844, 3.17653546942301702643588902945, 3.33620011226421082636492671267, 3.61187020248674200836956173386, 3.72523009581308654023372077230, 3.77309444083335479053955210645, 3.86153618672425635165382686946, 4.46345831168935850252696587765, 4.62378838901466027699821221400, 4.72943244324318017616117073121, 4.99084112716440111855460068245, 5.31807492551881003216459297810, 5.54542361702429454275172415652, 5.59759352229335320574216629958, 5.72514990553029491632028922693, 5.83337267646263694445845412637, 6.06361141705955454314960532953, 6.29906134716818663296225780485, 6.46661727348123593781891420818

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

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