Properties

Label 12-280e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.819\times 10^{14}$
Sign $1$
Analytic cond. $124.913$
Root an. cond. $1.49526$
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  − 3·5-s + 6·7-s − 3·11-s + 6·13-s − 6·17-s − 3·19-s − 3·23-s + 3·25-s − 12·27-s + 24·29-s − 12·31-s − 18·35-s − 9·37-s + 18·41-s + 24·43-s + 15·47-s + 12·49-s − 9·53-s + 9·55-s − 24·59-s + 6·61-s − 18·65-s − 6·67-s − 18·77-s + 18·79-s − 60·83-s + 18·85-s + ⋯
L(s)  = 1  − 1.34·5-s + 2.26·7-s − 0.904·11-s + 1.66·13-s − 1.45·17-s − 0.688·19-s − 0.625·23-s + 3/5·25-s − 2.30·27-s + 4.45·29-s − 2.15·31-s − 3.04·35-s − 1.47·37-s + 2.81·41-s + 3.65·43-s + 2.18·47-s + 12/7·49-s − 1.23·53-s + 1.21·55-s − 3.12·59-s + 0.768·61-s − 2.23·65-s − 0.733·67-s − 2.05·77-s + 2.02·79-s − 6.58·83-s + 1.95·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(124.913\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.451942030\)
\(L(\frac12)\) \(\approx\) \(1.451942030\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 - 6 T + 24 T^{2} - 80 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( ( 1 + 2 p T^{3} + p^{3} T^{6} )^{2} \)
11 \( 1 + 3 T - 17 T^{3} - 84 T^{4} - 3 p T^{5} + 106 p T^{6} - 3 p^{2} T^{7} - 84 p^{2} T^{8} - 17 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 3 T + 15 T^{2} - 10 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 + 3 T - 24 T^{2} - 161 T^{3} + 72 T^{4} + 1611 T^{5} + 6678 T^{6} + 1611 p T^{7} + 72 p^{2} T^{8} - 161 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T - 45 T^{2} - 128 T^{3} + 1239 T^{4} + 1965 T^{5} - 26470 T^{6} + 1965 p T^{7} + 1239 p^{2} T^{8} - 128 p^{3} T^{9} - 45 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 12 T + 39 T^{2} + 28 T^{3} + 378 T^{4} - 1908 T^{5} - 41121 T^{6} - 1908 p T^{7} + 378 p^{2} T^{8} + 28 p^{3} T^{9} + 39 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 30 T^{2} - 141 T^{3} + 3084 T^{4} + 45 p T^{5} - 140560 T^{6} + 45 p^{2} T^{7} + 3084 p^{2} T^{8} - 141 p^{3} T^{9} - 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 9 T + 78 T^{2} - 357 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 12 T + 168 T^{2} - 1054 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 15 T + 180 T^{2} - 1031 T^{3} + 2580 T^{4} + 38325 T^{5} - 352870 T^{6} + 38325 p T^{7} + 2580 p^{2} T^{8} - 1031 p^{3} T^{9} + 180 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 9 T - 6 T^{2} + 123 T^{3} - 12 T^{4} - 29007 T^{5} - 222032 T^{6} - 29007 p T^{7} - 12 p^{2} T^{8} + 123 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{3} \)
61 \( 1 - 6 T - 60 T^{2} - 200 T^{3} + 2232 T^{4} + 29898 T^{5} - 206826 T^{6} + 29898 p T^{7} + 2232 p^{2} T^{8} - 200 p^{3} T^{9} - 60 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 96 T^{2} - 832 T^{3} + 3708 T^{4} + 27990 T^{5} - 68142 T^{6} + 27990 p T^{7} + 3708 p^{2} T^{8} - 832 p^{3} T^{9} - 96 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + p T^{2} )^{6} \)
73 \( 1 - 111 T^{2} + 672 T^{3} + 4218 T^{4} - 37296 T^{5} - 8503 T^{6} - 37296 p T^{7} + 4218 p^{2} T^{8} + 672 p^{3} T^{9} - 111 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 18 T + 87 T^{2} - 114 T^{3} + 78 T^{4} + 882 p T^{5} - 1163581 T^{6} + 882 p^{2} T^{7} + 78 p^{2} T^{8} - 114 p^{3} T^{9} + 87 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 30 T + 540 T^{2} + 5884 T^{3} + 540 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 240 T^{2} + 84 T^{3} + 36240 T^{4} - 10080 T^{5} - 3714698 T^{6} - 10080 p T^{7} + 36240 p^{2} T^{8} + 84 p^{3} T^{9} - 240 p^{4} T^{10} + p^{6} T^{12} \)
97 \( ( 1 + 2 T + p T^{2} )^{6} \)
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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

−6.62543321457133270199307982323, −6.16521403520588153953849409205, −6.02983673712212829535278577653, −5.75181646687244594181674559072, −5.73830927318164739903995169283, −5.73394011786525625886312420609, −5.59150258073620404340717824829, −4.98945116442219325126059479041, −4.76431215049530286584086950921, −4.71358651196764302376313780721, −4.43074120077593723920974351735, −4.41664561998213504208498772708, −4.20934300863081578036029324396, −4.09076195870915743930351395785, −3.64197973225146673299535902780, −3.62295099703294338690351473884, −3.37664838505248924435401190206, −2.68431843027983915432498646441, −2.68089200758246309366403551718, −2.47305310117387144896556057520, −2.29818703879983319749571464018, −1.65957801706493661487406782341, −1.44117665930974068488389560900, −1.26543248979319791345994143738, −0.43678025571468972624121645034, 0.43678025571468972624121645034, 1.26543248979319791345994143738, 1.44117665930974068488389560900, 1.65957801706493661487406782341, 2.29818703879983319749571464018, 2.47305310117387144896556057520, 2.68089200758246309366403551718, 2.68431843027983915432498646441, 3.37664838505248924435401190206, 3.62295099703294338690351473884, 3.64197973225146673299535902780, 4.09076195870915743930351395785, 4.20934300863081578036029324396, 4.41664561998213504208498772708, 4.43074120077593723920974351735, 4.71358651196764302376313780721, 4.76431215049530286584086950921, 4.98945116442219325126059479041, 5.59150258073620404340717824829, 5.73394011786525625886312420609, 5.73830927318164739903995169283, 5.75181646687244594181674559072, 6.02983673712212829535278577653, 6.16521403520588153953849409205, 6.62543321457133270199307982323

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

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