Properties

Label 12-5070e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.698\times 10^{22}$
Sign $1$
Analytic cond. $4.40261\times 10^{9}$
Root an. cond. $6.36271$
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  − 6·3-s − 3·4-s + 21·9-s + 18·12-s + 6·16-s + 18·17-s + 2·23-s − 3·25-s − 56·27-s − 8·29-s − 63·36-s + 18·43-s − 36·48-s + 16·49-s − 108·51-s − 34·53-s + 2·61-s − 10·64-s − 54·68-s − 12·69-s + 18·75-s − 6·79-s + 126·81-s + 48·87-s − 6·92-s + 9·100-s − 36·101-s + ⋯
L(s)  = 1  − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s + 3/2·16-s + 4.36·17-s + 0.417·23-s − 3/5·25-s − 10.7·27-s − 1.48·29-s − 10.5·36-s + 2.74·43-s − 5.19·48-s + 16/7·49-s − 15.1·51-s − 4.67·53-s + 0.256·61-s − 5/4·64-s − 6.54·68-s − 1.44·69-s + 2.07·75-s − 0.675·79-s + 14·81-s + 5.14·87-s − 0.625·92-s + 9/10·100-s − 3.58·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.40261\times 10^{9}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1027617990\)
\(L(\frac12)\) \(\approx\) \(0.1027617990\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
13 \( 1 \)
good7 \( 1 - 16 T^{2} + 160 T^{4} - 1189 T^{6} + 160 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 32 T^{2} + 60 p T^{4} - 8597 T^{6} + 60 p^{3} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 9 T + 57 T^{2} - 263 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 81 T^{2} + 3137 T^{4} - 74273 T^{6} + 3137 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 - T + 67 T^{2} - 45 T^{3} + 67 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 4 T + 20 T^{2} - 7 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 152 T^{2} + 10372 T^{4} - 410333 T^{6} + 10372 p^{2} T^{8} - 152 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 184 T^{2} + 15292 T^{4} - 728281 T^{6} + 15292 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 233 T^{2} + 23109 T^{4} - 1249433 T^{6} + 23109 p^{2} T^{8} - 233 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 9 T + 107 T^{2} - 703 T^{3} + 107 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 79 T^{2} + 5310 T^{4} - 349483 T^{6} + 5310 p^{2} T^{8} - 79 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 17 T + 239 T^{2} + 1873 T^{3} + 239 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 71 T^{2} + 10665 T^{4} + 477095 T^{6} + 10665 p^{2} T^{8} + 71 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - T + 83 T^{2} - 9 T^{3} + 83 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 133 T^{2} + 11129 T^{4} - 870177 T^{6} + 11129 p^{2} T^{8} - 133 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 153 T^{2} + 17193 T^{4} - 1532313 T^{6} + 17193 p^{2} T^{8} - 153 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 389 T^{2} + 66313 T^{4} - 6311369 T^{6} + 66313 p^{2} T^{8} - 389 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 3 T + 93 T^{2} + 825 T^{3} + 93 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 88 T^{2} + 14916 T^{4} + 704515 T^{6} + 14916 p^{2} T^{8} + 88 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 213 T^{2} + 23129 T^{4} - 2123993 T^{6} + 23129 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 264 T^{2} + 26252 T^{4} - 1921721 T^{6} + 26252 p^{2} T^{8} - 264 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.26752923216014626568798276477, −4.25518561920276138472726653177, −4.17991788459464686052686484190, −3.98354821229135431381931697506, −3.73280408688467849609473950728, −3.56334698325609932847959849054, −3.31203156791596538353386672196, −3.30065109062583983467826223876, −3.23604144212047327133891471445, −3.08428402525564727233883323644, −2.85484740136735737699047097159, −2.78130647484385035095795059154, −2.32534325939617105434455657022, −2.27432716196676775544224912232, −2.03279012329550187144364909202, −1.82552605607604744689516870748, −1.61882715963952527658195957855, −1.46210012705759021330957077697, −1.26677238757252230138273414988, −1.03087951186062140500447256184, −0.999108111842300695871210563058, −0.932946371239093848490886703872, −0.67139001184784154617771833516, −0.25856966155623898202212340468, −0.088394300558686218319412513190, 0.088394300558686218319412513190, 0.25856966155623898202212340468, 0.67139001184784154617771833516, 0.932946371239093848490886703872, 0.999108111842300695871210563058, 1.03087951186062140500447256184, 1.26677238757252230138273414988, 1.46210012705759021330957077697, 1.61882715963952527658195957855, 1.82552605607604744689516870748, 2.03279012329550187144364909202, 2.27432716196676775544224912232, 2.32534325939617105434455657022, 2.78130647484385035095795059154, 2.85484740136735737699047097159, 3.08428402525564727233883323644, 3.23604144212047327133891471445, 3.30065109062583983467826223876, 3.31203156791596538353386672196, 3.56334698325609932847959849054, 3.73280408688467849609473950728, 3.98354821229135431381931697506, 4.17991788459464686052686484190, 4.25518561920276138472726653177, 4.26752923216014626568798276477

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

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