Properties

Label 12-2016e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.713\times 10^{19}$
Sign $1$
Analytic cond. $1.74022\times 10^{7}$
Root an. cond. $4.01221$
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  + 3·7-s − 6·13-s + 6·17-s + 3·19-s + 6·23-s + 6·25-s − 24·29-s + 3·31-s − 3·37-s + 12·41-s − 30·43-s − 12·47-s + 9·49-s + 6·53-s − 12·59-s + 18·61-s + 9·67-s − 33·73-s − 27·79-s + 36·83-s − 12·89-s − 18·91-s + 18·101-s + 15·103-s − 24·107-s − 9·109-s − 48·113-s + ⋯
L(s)  = 1  + 1.13·7-s − 1.66·13-s + 1.45·17-s + 0.688·19-s + 1.25·23-s + 6/5·25-s − 4.45·29-s + 0.538·31-s − 0.493·37-s + 1.87·41-s − 4.57·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s + 2.30·61-s + 1.09·67-s − 3.86·73-s − 3.03·79-s + 3.95·83-s − 1.27·89-s − 1.88·91-s + 1.79·101-s + 1.47·103-s − 2.32·107-s − 0.862·109-s − 4.51·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.74022\times 10^{7}\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7613872774\)
\(L(\frac12)\) \(\approx\) \(0.7613872774\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 3 T + 5 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 24 p T^{7} + 6 p^{2} T^{8} - 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 6 T^{2} + 76 T^{3} - 30 T^{4} - 228 T^{5} + 3710 T^{6} - 228 p T^{7} - 30 p^{2} T^{8} + 76 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 6 T + 9 T^{2} + 54 T^{3} - 378 T^{4} + 858 T^{5} - 1307 T^{6} + 858 p T^{7} - 378 p^{2} T^{8} + 54 p^{3} T^{9} + 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 12 T^{2} - 59 T^{3} + 36 T^{4} + 1269 T^{5} + 2094 T^{6} + 1269 p T^{7} + 36 p^{2} T^{8} - 59 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T - 9 T^{2} + 90 T^{3} - 90 T^{4} + 1146 T^{5} - 8885 T^{6} + 1146 p T^{7} - 90 p^{2} T^{8} + 90 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 12 T + 126 T^{2} + 728 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 3 T - 63 T^{2} + 2 p T^{3} + 2535 T^{4} + 501 T^{5} - 90450 T^{6} + 501 p T^{7} + 2535 p^{2} T^{8} + 2 p^{4} T^{9} - 63 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T - 18 T^{2} + 373 T^{3} + 168 T^{4} - 5829 T^{5} + 83328 T^{6} - 5829 p T^{7} + 168 p^{2} T^{8} + 373 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 6 T + 27 T^{2} + 20 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 15 T + 177 T^{2} + 1318 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 12 T - 9 T^{2} - 196 T^{3} + 4026 T^{4} + 9372 T^{5} - 178417 T^{6} + 9372 p T^{7} + 4026 p^{2} T^{8} - 196 p^{3} T^{9} - 9 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 42 T^{2} + 1136 T^{3} - 54 p T^{4} - 29802 T^{5} + 517382 T^{6} - 29802 p T^{7} - 54 p^{3} T^{8} + 1136 p^{3} T^{9} - 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 12 T - 54 T^{2} - 292 T^{3} + 11514 T^{4} + 25536 T^{5} - 623986 T^{6} + 25536 p T^{7} + 11514 p^{2} T^{8} - 292 p^{3} T^{9} - 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 129 T^{2} - 222 T^{3} - 3462 T^{4} + 40662 T^{5} - 368431 T^{6} + 40662 p T^{7} - 3462 p^{2} T^{8} - 222 p^{3} T^{9} + 129 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 108 T^{2} + 383 T^{3} + 13680 T^{4} - 9405 T^{5} - 1069626 T^{6} - 9405 p T^{7} + 13680 p^{2} T^{8} + 383 p^{3} T^{9} - 108 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6} )^{2} \)
73 \( 1 + 33 T + 534 T^{2} + 6687 T^{3} + 75648 T^{4} + 728637 T^{5} + 6291524 T^{6} + 728637 p T^{7} + 75648 p^{2} T^{8} + 6687 p^{3} T^{9} + 534 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 297 T^{2} + 2426 T^{3} + 27783 T^{4} + 321003 T^{5} + 3028254 T^{6} + 321003 p T^{7} + 27783 p^{2} T^{8} + 2426 p^{3} T^{9} + 297 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 18 T + 234 T^{2} - 2040 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 12 T - 135 T^{2} - 700 T^{3} + 28722 T^{4} + 63804 T^{5} - 2561959 T^{6} + 63804 p T^{7} + 28722 p^{2} T^{8} - 700 p^{3} T^{9} - 135 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} )^{2} \)
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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

−5.04653300330250447935068583154, −4.63537422366507052260574155308, −4.43991440117219630056343807179, −4.32121741398335299155693880133, −4.27737710529245428659263649229, −3.95189276788748560077283383409, −3.81944258249612148177721423255, −3.62944877235138402409377989322, −3.54742151830888214202155914206, −3.33338459293055502139585648331, −3.18708568970907513516538584913, −3.18344824037263103341780062431, −2.88650981676102058018954180813, −2.59739539333473875160493946582, −2.36837212295496969880612972333, −2.35169671312759799069806722249, −2.31190695290768518627413172043, −1.90010370287928665363890575345, −1.69579144208649132942015013121, −1.39320632764103230266074089697, −1.22206174304749980329296937364, −1.19830354541306160240875287771, −1.18524722282997523842063034807, −0.26386909966662912342689837825, −0.17682678000027265359468528391, 0.17682678000027265359468528391, 0.26386909966662912342689837825, 1.18524722282997523842063034807, 1.19830354541306160240875287771, 1.22206174304749980329296937364, 1.39320632764103230266074089697, 1.69579144208649132942015013121, 1.90010370287928665363890575345, 2.31190695290768518627413172043, 2.35169671312759799069806722249, 2.36837212295496969880612972333, 2.59739539333473875160493946582, 2.88650981676102058018954180813, 3.18344824037263103341780062431, 3.18708568970907513516538584913, 3.33338459293055502139585648331, 3.54742151830888214202155914206, 3.62944877235138402409377989322, 3.81944258249612148177721423255, 3.95189276788748560077283383409, 4.27737710529245428659263649229, 4.32121741398335299155693880133, 4.43991440117219630056343807179, 4.63537422366507052260574155308, 5.04653300330250447935068583154

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

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