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Properties

Label 12-6422e6-1.1-c1e6-0-0
Degree $12$
Conductor $7.015\times 10^{22}$
Sign $1$
Analytic cond. $1.81837\times 10^{10}$
Root an. cond. $7.16100$
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  − 6·2-s + 2·3-s + 21·4-s − 2·5-s − 12·6-s − 7-s − 56·8-s − 2·9-s + 12·10-s − 2·11-s + 42·12-s + 6·14-s − 4·15-s + 126·16-s − 2·17-s + 12·18-s + 6·19-s − 42·20-s − 2·21-s + 12·22-s + 8·23-s − 112·24-s − 5·25-s − 10·27-s − 21·28-s + 20·29-s + 24·30-s + ⋯
L(s)  = 1  − 4.24·2-s + 1.15·3-s + 21/2·4-s − 0.894·5-s − 4.89·6-s − 0.377·7-s − 19.7·8-s − 2/3·9-s + 3.79·10-s − 0.603·11-s + 12.1·12-s + 1.60·14-s − 1.03·15-s + 63/2·16-s − 0.485·17-s + 2.82·18-s + 1.37·19-s − 9.39·20-s − 0.436·21-s + 2.55·22-s + 1.66·23-s − 22.8·24-s − 25-s − 1.92·27-s − 3.96·28-s + 3.71·29-s + 4.38·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 13^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.81837\times 10^{10}\)
Root analytic conductor: \(7.16100\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 13^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9991987720\)
\(L(\frac12)\) \(\approx\) \(0.9991987720\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
13 \( 1 \)
19 \( ( 1 - T )^{6} \)
good3 \( 1 - 2 T + 2 p T^{2} - 2 p T^{3} + 4 T^{4} + 4 p T^{5} - 25 T^{6} + 4 p^{2} T^{7} + 4 p^{2} T^{8} - 2 p^{4} T^{9} + 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 2 T + 9 T^{2} - T^{3} + 19 T^{4} - 121 T^{5} - 74 T^{6} - 121 p T^{7} + 19 p^{2} T^{8} - p^{3} T^{9} + 9 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + T + 25 T^{2} + 10 T^{3} + 316 T^{4} + 82 T^{5} + 2711 T^{6} + 82 p T^{7} + 316 p^{2} T^{8} + 10 p^{3} T^{9} + 25 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T + 45 T^{2} + 59 T^{3} + 955 T^{4} + 881 T^{5} + 12718 T^{6} + 881 p T^{7} + 955 p^{2} T^{8} + 59 p^{3} T^{9} + 45 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 23 T^{2} + 121 T^{3} + 577 T^{4} + 1913 T^{5} + 16270 T^{6} + 1913 p T^{7} + 577 p^{2} T^{8} + 121 p^{3} T^{9} + 23 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 8 T + 122 T^{2} - 768 T^{3} + 6470 T^{4} - 32106 T^{5} + 192741 T^{6} - 32106 p T^{7} + 6470 p^{2} T^{8} - 768 p^{3} T^{9} + 122 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 20 T + 316 T^{2} - 3346 T^{3} + 29692 T^{4} - 207276 T^{5} + 1239439 T^{6} - 207276 p T^{7} + 29692 p^{2} T^{8} - 3346 p^{3} T^{9} + 316 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 86 T^{2} + 134 T^{3} + 4481 T^{4} + 4319 T^{5} + 155736 T^{6} + 4319 p T^{7} + 4481 p^{2} T^{8} + 134 p^{3} T^{9} + 86 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 9 T + 125 T^{2} - 924 T^{3} + 9203 T^{4} - 54235 T^{5} + 396982 T^{6} - 54235 p T^{7} + 9203 p^{2} T^{8} - 924 p^{3} T^{9} + 125 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 146 T^{2} + 338 T^{3} + 10729 T^{4} + 18647 T^{5} + 526472 T^{6} + 18647 p T^{7} + 10729 p^{2} T^{8} + 338 p^{3} T^{9} + 146 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 13 T + 277 T^{2} - 2558 T^{3} + 30395 T^{4} - 209705 T^{5} + 1748838 T^{6} - 209705 p T^{7} + 30395 p^{2} T^{8} - 2558 p^{3} T^{9} + 277 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 20 T + 426 T^{2} + 5117 T^{3} + 60433 T^{4} + 499748 T^{5} + 4005361 T^{6} + 499748 p T^{7} + 60433 p^{2} T^{8} + 5117 p^{3} T^{9} + 426 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 25 T + 397 T^{2} - 4292 T^{3} + 35242 T^{4} - 248784 T^{5} + 1703239 T^{6} - 248784 p T^{7} + 35242 p^{2} T^{8} - 4292 p^{3} T^{9} + 397 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 134 T^{2} - 578 T^{3} + 12478 T^{4} - 41946 T^{5} + 992767 T^{6} - 41946 p T^{7} + 12478 p^{2} T^{8} - 578 p^{3} T^{9} + 134 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 6 T + 131 T^{2} - 87 T^{3} + 10769 T^{4} - 25075 T^{5} + 977014 T^{6} - 25075 p T^{7} + 10769 p^{2} T^{8} - 87 p^{3} T^{9} + 131 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 32 T + 645 T^{2} - 9089 T^{3} + 107061 T^{4} - 1058297 T^{5} + 9329378 T^{6} - 1058297 p T^{7} + 107061 p^{2} T^{8} - 9089 p^{3} T^{9} + 645 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 39 T + 959 T^{2} + 16342 T^{3} + 221869 T^{4} + 2428363 T^{5} + 22436510 T^{6} + 2428363 p T^{7} + 221869 p^{2} T^{8} + 16342 p^{3} T^{9} + 959 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 7 T + 322 T^{2} + 2416 T^{3} + 49189 T^{4} + 335965 T^{5} + 4530785 T^{6} + 335965 p T^{7} + 49189 p^{2} T^{8} + 2416 p^{3} T^{9} + 322 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 18 T + 413 T^{2} - 3901 T^{3} + 49289 T^{4} - 284671 T^{5} + 3531642 T^{6} - 284671 p T^{7} + 49289 p^{2} T^{8} - 3901 p^{3} T^{9} + 413 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 7 T + 232 T^{2} - 2052 T^{3} + 35075 T^{4} - 259765 T^{5} + 3676768 T^{6} - 259765 p T^{7} + 35075 p^{2} T^{8} - 2052 p^{3} T^{9} + 232 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T + 428 T^{2} + 4030 T^{3} + 81745 T^{4} + 719673 T^{5} + 9179524 T^{6} + 719673 p T^{7} + 81745 p^{2} T^{8} + 4030 p^{3} T^{9} + 428 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 30 T^{2} + 1692 T^{3} - p T^{4} - 26392 T^{5} + 2392900 T^{6} - 26392 p T^{7} - p^{3} T^{8} + 1692 p^{3} T^{9} + 30 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.02320045834745400911994232610, −3.68206145026329823932668471157, −3.58166820262695831447832574183, −3.41571980770762566232337907931, −3.36759677764893899810684074500, −3.35798693615355757646291505011, −3.25402166071145597725352964728, −2.80461805608176712481118442454, −2.77958079915483790586646493573, −2.72251831891855577316160129366, −2.69989467304706024071831885157, −2.65092322904080864291421164873, −2.31025574627722436287864088666, −2.05067636241194457967742106596, −2.01745549854128739105736967595, −1.92981396758046487443205283748, −1.67438768644457395835364167935, −1.57626681012703546414680490583, −1.19897412998178447944003785308, −1.01077085928085216478211767068, −0.951832576274682039102356258650, −0.822303508562835408245275805476, −0.47855464478010743871745850246, −0.41027443456825437664075760605, −0.27729597412572189533102691403, 0.27729597412572189533102691403, 0.41027443456825437664075760605, 0.47855464478010743871745850246, 0.822303508562835408245275805476, 0.951832576274682039102356258650, 1.01077085928085216478211767068, 1.19897412998178447944003785308, 1.57626681012703546414680490583, 1.67438768644457395835364167935, 1.92981396758046487443205283748, 2.01745549854128739105736967595, 2.05067636241194457967742106596, 2.31025574627722436287864088666, 2.65092322904080864291421164873, 2.69989467304706024071831885157, 2.72251831891855577316160129366, 2.77958079915483790586646493573, 2.80461805608176712481118442454, 3.25402166071145597725352964728, 3.35798693615355757646291505011, 3.36759677764893899810684074500, 3.41571980770762566232337907931, 3.58166820262695831447832574183, 3.68206145026329823932668471157, 4.02320045834745400911994232610

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

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