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Properties

Label 12-570e6-1.1-c1e6-0-9
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  + 3·7-s + 8-s + 9·11-s − 3·13-s − 3·17-s + 9·19-s − 3·23-s + 27-s + 33·29-s + 18·37-s − 6·41-s − 15·43-s + 18·47-s + 21·49-s + 12·53-s + 3·56-s − 9·59-s + 21·61-s − 24·67-s − 42·71-s − 45·73-s + 27·77-s + 9·79-s − 9·83-s + 9·88-s − 21·89-s − 9·91-s + ⋯
L(s)  = 1  + 1.13·7-s + 0.353·8-s + 2.71·11-s − 0.832·13-s − 0.727·17-s + 2.06·19-s − 0.625·23-s + 0.192·27-s + 6.12·29-s + 2.95·37-s − 0.937·41-s − 2.28·43-s + 2.62·47-s + 3·49-s + 1.64·53-s + 0.400·56-s − 1.17·59-s + 2.68·61-s − 2.93·67-s − 4.98·71-s − 5.26·73-s + 3.07·77-s + 1.01·79-s − 0.987·83-s + 0.959·88-s − 2.22·89-s − 0.943·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.973030838\)
\(L(\frac12)\) \(\approx\) \(8.973030838\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 - T^{3} + T^{6} \)
5 \( 1 + T^{3} + T^{6} \)
19 \( 1 - 9 T + 72 T^{2} - 341 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 18 p T^{5} - 1161 T^{6} - 18 p^{2} T^{7} + 171 p^{2} T^{8} + 19 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 9 T + 24 T^{2} - 83 T^{3} + 687 T^{4} - 2058 T^{5} + 3347 T^{6} - 2058 p T^{7} + 687 p^{2} T^{8} - 83 p^{3} T^{9} + 24 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T - 18 T^{2} - 132 T^{3} - 207 T^{4} + 1101 T^{5} + 8009 T^{6} + 1101 p T^{7} - 207 p^{2} T^{8} - 132 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T - 6 T^{2} - 74 T^{3} - 267 T^{4} + 801 T^{5} + 9869 T^{6} + 801 p T^{7} - 267 p^{2} T^{8} - 74 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 3 T^{2} + 61 T^{3} - 132 T^{4} - 2970 T^{5} - 12235 T^{6} - 2970 p T^{7} - 132 p^{2} T^{8} + 61 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 33 T + 477 T^{2} - 3951 T^{3} + 20574 T^{4} - 73068 T^{5} + 275545 T^{6} - 73068 p T^{7} + 20574 p^{2} T^{8} - 3951 p^{3} T^{9} + 477 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 30 T^{2} + 18 T^{3} - 30 T^{4} - 270 T^{5} + 31763 T^{6} - 270 p T^{7} - 30 p^{2} T^{8} + 18 p^{3} T^{9} - 30 p^{4} T^{10} + p^{6} T^{12} \)
37 \( ( 1 - 9 T + 81 T^{2} - 415 T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 6 T - 12 T^{2} - 467 T^{3} - 1785 T^{4} + 11871 T^{5} + 131933 T^{6} + 11871 p T^{7} - 1785 p^{2} T^{8} - 467 p^{3} T^{9} - 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 180 T^{2} + 1878 T^{3} + 15948 T^{4} + 127131 T^{5} + 895085 T^{6} + 127131 p T^{7} + 15948 p^{2} T^{8} + 1878 p^{3} T^{9} + 180 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 144 T^{2} - 761 T^{3} + 1953 T^{4} + 18765 T^{5} - 248923 T^{6} + 18765 p T^{7} + 1953 p^{2} T^{8} - 761 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 12 T + 57 T^{2} + 83 T^{3} - 39 p T^{4} + 15021 T^{5} - 57406 T^{6} + 15021 p T^{7} - 39 p^{3} T^{8} + 83 p^{3} T^{9} + 57 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 9 T - 108 T^{2} - 950 T^{3} + 4860 T^{4} + 25605 T^{5} - 217747 T^{6} + 25605 p T^{7} + 4860 p^{2} T^{8} - 950 p^{3} T^{9} - 108 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 21 T + 174 T^{2} - 346 T^{3} - 6255 T^{4} + 101205 T^{5} - 948315 T^{6} + 101205 p T^{7} - 6255 p^{2} T^{8} - 346 p^{3} T^{9} + 174 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 24 T + 270 T^{2} + 2121 T^{3} + 16119 T^{4} + 119211 T^{5} + 886265 T^{6} + 119211 p T^{7} + 16119 p^{2} T^{8} + 2121 p^{3} T^{9} + 270 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 42 T + 768 T^{2} + 6995 T^{3} + 8625 T^{4} - 607455 T^{5} - 7981123 T^{6} - 607455 p T^{7} + 8625 p^{2} T^{8} + 6995 p^{3} T^{9} + 768 p^{4} T^{10} + 42 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 45 T + 990 T^{2} + 14830 T^{3} + 180225 T^{4} + 1920375 T^{5} + 17772933 T^{6} + 1920375 p T^{7} + 180225 p^{2} T^{8} + 14830 p^{3} T^{9} + 990 p^{4} T^{10} + 45 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T - 9 T^{2} + 1285 T^{3} - 8802 T^{4} - 25326 T^{5} + 878817 T^{6} - 25326 p T^{7} - 8802 p^{2} T^{8} + 1285 p^{3} T^{9} - 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 9 T - 138 T^{2} - 943 T^{3} + 16671 T^{4} + 53898 T^{5} - 1272349 T^{6} + 53898 p T^{7} + 16671 p^{2} T^{8} - 943 p^{3} T^{9} - 138 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 21 T + 336 T^{2} + 3598 T^{3} + 46200 T^{4} + 492093 T^{5} + 5422775 T^{6} + 492093 p T^{7} + 46200 p^{2} T^{8} + 3598 p^{3} T^{9} + 336 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T + 120 T^{2} + 662 T^{3} + 11214 T^{4} - 15462 T^{5} + 488391 T^{6} - 15462 p T^{7} + 11214 p^{2} T^{8} + 662 p^{3} T^{9} + 120 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.89162103857437326110285938246, −5.71083132613456088364277271334, −5.51413733043928893390290358428, −5.31048787589344637787960122200, −4.86522422544245164776163218727, −4.64938655325479486607847845225, −4.56990773945604073961389378896, −4.51196378798737725531282555328, −4.45328148994919459670860815343, −4.33703696528672797625567549490, −4.13750692796181119544317423207, −4.05969695575879990016488932543, −3.43489908611881342465906104312, −3.23426750902679959120851634216, −3.12285594160893025226368868776, −3.08106771429625765471144795738, −2.69274044259346828405581098650, −2.60134949376488927445693272143, −2.36644933394528704446493398275, −1.81992389596498091830316286085, −1.80810692912743764914209796684, −1.30008748831304633294229346016, −1.20613991301633819355861263197, −0.842928225827578439025651852914, −0.794550516355902460409054775725, 0.794550516355902460409054775725, 0.842928225827578439025651852914, 1.20613991301633819355861263197, 1.30008748831304633294229346016, 1.80810692912743764914209796684, 1.81992389596498091830316286085, 2.36644933394528704446493398275, 2.60134949376488927445693272143, 2.69274044259346828405581098650, 3.08106771429625765471144795738, 3.12285594160893025226368868776, 3.23426750902679959120851634216, 3.43489908611881342465906104312, 4.05969695575879990016488932543, 4.13750692796181119544317423207, 4.33703696528672797625567549490, 4.45328148994919459670860815343, 4.51196378798737725531282555328, 4.56990773945604073961389378896, 4.64938655325479486607847845225, 4.86522422544245164776163218727, 5.31048787589344637787960122200, 5.51413733043928893390290358428, 5.71083132613456088364277271334, 5.89162103857437326110285938246

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

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