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Properties

Label 12-670e6-1.1-c1e6-0-0
Degree $12$
Conductor $9.046\times 10^{16}$
Sign $1$
Analytic cond. $23448.3$
Root an. cond. $2.31300$
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 − 2·3-s + 3·4-s + 6·5-s − 6·6-s − 2·7-s − 2·8-s − 5·9-s + 18·10-s + 8·11-s − 6·12-s − 2·13-s − 6·14-s − 12·15-s − 9·16-s + 3·17-s − 15·18-s + 3·19-s + 18·20-s + 4·21-s + 24·22-s + 4·23-s + 4·24-s + 21·25-s − 6·26-s + 14·27-s − 6·28-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.15·3-s + 3/2·4-s + 2.68·5-s − 2.44·6-s − 0.755·7-s − 0.707·8-s − 5/3·9-s + 5.69·10-s + 2.41·11-s − 1.73·12-s − 0.554·13-s − 1.60·14-s − 3.09·15-s − 9/4·16-s + 0.727·17-s − 3.53·18-s + 0.688·19-s + 4.02·20-s + 0.872·21-s + 5.11·22-s + 0.834·23-s + 0.816·24-s + 21/5·25-s − 1.17·26-s + 2.69·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 67^{6}\)
Sign: $1$
Analytic conductor: \(23448.3\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 67^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.687798153\)
\(L(\frac12)\) \(\approx\) \(3.687798153\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
67 \( 1 + 18 T + 225 T^{2} + 1960 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( ( 1 + T + 4 T^{2} + p T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 + 2 T - p T^{2} - 30 T^{3} - 8 T^{4} + 76 T^{5} + 319 T^{6} + 76 p T^{7} - 8 p^{2} T^{8} - 30 p^{3} T^{9} - p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 19 T^{2} - 32 T^{3} + 218 T^{4} - 428 T^{5} - 461 T^{6} - 428 p T^{7} + 218 p^{2} T^{8} - 32 p^{3} T^{9} + 19 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T - 25 T^{2} - 42 T^{3} + 28 p T^{4} + 280 T^{5} - 4715 T^{6} + 280 p T^{7} + 28 p^{3} T^{8} - 42 p^{3} T^{9} - 25 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T - 27 T^{2} + 114 T^{3} + 351 T^{4} - 1191 T^{5} - 2486 T^{6} - 1191 p T^{7} + 351 p^{2} T^{8} + 114 p^{3} T^{9} - 27 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 - 8 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \)
23 \( 1 - 4 T - 47 T^{2} + 80 T^{3} + 1856 T^{4} - 982 T^{5} - 47861 T^{6} - 982 p T^{7} + 1856 p^{2} T^{8} + 80 p^{3} T^{9} - 47 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T - 27 T^{2} - 354 T^{3} - 54 T^{4} + 6180 T^{5} + 37357 T^{6} + 6180 p T^{7} - 54 p^{2} T^{8} - 354 p^{3} T^{9} - 27 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 16 T + 99 T^{2} + 432 T^{3} + 3142 T^{4} + 22912 T^{5} + 131047 T^{6} + 22912 p T^{7} + 3142 p^{2} T^{8} + 432 p^{3} T^{9} + 99 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 10 T + 105 T^{2} + 726 T^{3} + 4090 T^{4} + 19330 T^{5} + 114025 T^{6} + 19330 p T^{7} + 4090 p^{2} T^{8} + 726 p^{3} T^{9} + 105 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 10 T - 47 T^{2} - 194 T^{3} + 7262 T^{4} + 18574 T^{5} - 224003 T^{6} + 18574 p T^{7} + 7262 p^{2} T^{8} - 194 p^{3} T^{9} - 47 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - T + 124 T^{2} - 83 T^{3} + 124 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \)
53 \( ( 1 - 12 T + 189 T^{2} - 1290 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( ( 1 + T + 96 T^{2} + 361 T^{3} + 96 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 - 14 T - 15 T^{2} + 414 T^{3} + 8074 T^{4} - 47030 T^{5} - 155279 T^{6} - 47030 p T^{7} + 8074 p^{2} T^{8} + 414 p^{3} T^{9} - 15 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 22 T + 115 T^{2} - 1162 T^{3} + 33446 T^{4} - 241528 T^{5} + 705031 T^{6} - 241528 p T^{7} + 33446 p^{2} T^{8} - 1162 p^{3} T^{9} + 115 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 3 T - 195 T^{2} - 178 T^{3} + 25539 T^{4} + 7215 T^{5} - 2151870 T^{6} + 7215 p T^{7} + 25539 p^{2} T^{8} - 178 p^{3} T^{9} - 195 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 12 T - 45 T^{2} + 1460 T^{3} - 1578 T^{4} - 68844 T^{5} + 739911 T^{6} - 68844 p T^{7} - 1578 p^{2} T^{8} + 1460 p^{3} T^{9} - 45 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 10 T - 101 T^{2} + 1382 T^{3} + 6758 T^{4} - 76126 T^{5} - 22193 T^{6} - 76126 p T^{7} + 6758 p^{2} T^{8} + 1382 p^{3} T^{9} - 101 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 3 T + 198 T^{2} + 255 T^{3} + 198 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 17 T - 31 T^{2} - 1074 T^{3} + 15991 T^{4} + 146833 T^{5} - 218606 T^{6} + 146833 p T^{7} + 15991 p^{2} T^{8} - 1074 p^{3} T^{9} - 31 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.59802419615995192348886035554, −5.51766782395797704190411070164, −5.43014077171549889497946358913, −5.14594883614172097511375128260, −5.11154929521670251157667995002, −4.96358356936324905597102788360, −4.75828445548160373779705997640, −4.47481322137972208428962411716, −4.27530377402104347160313070222, −4.00775858484400262491542643551, −3.89151888991004303710072380311, −3.57488234180378721836734137404, −3.39316063216679364977496432433, −3.30882509336451798676502847616, −3.30173385619897281444680134255, −3.19126471747173875784542296364, −2.55813269034314397415309353503, −2.47884610247343454185732865652, −2.33726156535412897149834713278, −2.03932702877338548570707409551, −1.74105711486051530390101199662, −1.62714314150999692156941851691, −1.11153252346691064056518930275, −0.893467139065450522110191279784, −0.25874047466835349840195160082, 0.25874047466835349840195160082, 0.893467139065450522110191279784, 1.11153252346691064056518930275, 1.62714314150999692156941851691, 1.74105711486051530390101199662, 2.03932702877338548570707409551, 2.33726156535412897149834713278, 2.47884610247343454185732865652, 2.55813269034314397415309353503, 3.19126471747173875784542296364, 3.30173385619897281444680134255, 3.30882509336451798676502847616, 3.39316063216679364977496432433, 3.57488234180378721836734137404, 3.89151888991004303710072380311, 4.00775858484400262491542643551, 4.27530377402104347160313070222, 4.47481322137972208428962411716, 4.75828445548160373779705997640, 4.96358356936324905597102788360, 5.11154929521670251157667995002, 5.14594883614172097511375128260, 5.43014077171549889497946358913, 5.51766782395797704190411070164, 5.59802419615995192348886035554

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

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