Properties

Label 12-75e12-1.1-c1e6-0-3
Degree $12$
Conductor $3.168\times 10^{22}$
Sign $1$
Analytic cond. $8.21103\times 10^{9}$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·7-s − 8-s + 2·14-s − 2·17-s − 2·19-s + 23-s − 31·29-s − 2·31-s − 3·32-s + 2·34-s − 22·37-s + 2·38-s − 33·41-s − 3·43-s − 46-s + 6·47-s − 17·49-s − 14·53-s + 2·56-s + 31·58-s + 8·59-s + 34·61-s + 2·62-s + 64-s + 2·67-s + 3·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.755·7-s − 0.353·8-s + 0.534·14-s − 0.485·17-s − 0.458·19-s + 0.208·23-s − 5.75·29-s − 0.359·31-s − 0.530·32-s + 0.342·34-s − 3.61·37-s + 0.324·38-s − 5.15·41-s − 0.457·43-s − 0.147·46-s + 0.875·47-s − 2.42·49-s − 1.92·53-s + 0.267·56-s + 4.07·58-s + 1.04·59-s + 4.35·61-s + 0.254·62-s + 1/8·64-s + 0.244·67-s + 0.356·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(8.21103\times 10^{9}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{12} \cdot 5^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + T + T^{2} + p T^{3} + 3 T^{4} + 7 T^{5} + 11 T^{6} + 7 p T^{7} + 3 p^{2} T^{8} + p^{4} T^{9} + p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 3 p T^{2} + 4 p T^{3} + 248 T^{4} + 258 T^{5} + 2000 T^{6} + 258 p T^{7} + 248 p^{2} T^{8} + 4 p^{4} T^{9} + 3 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 37 T^{2} + 8 T^{3} + 723 T^{4} + 168 T^{5} + 9470 T^{6} + 168 p T^{7} + 723 p^{2} T^{8} + 8 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 22 T^{2} + 74 T^{3} + 447 T^{4} + 874 T^{5} + 8929 T^{6} + 874 p T^{7} + 447 p^{2} T^{8} + 74 p^{3} T^{9} + 22 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 2 T + 47 T^{2} + 90 T^{3} + 75 p T^{4} + 2852 T^{5} + 25434 T^{6} + 2852 p T^{7} + 75 p^{3} T^{8} + 90 p^{3} T^{9} + 47 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 40 T^{2} + 70 T^{3} + 1201 T^{4} + 1530 T^{5} + 24751 T^{6} + 1530 p T^{7} + 1201 p^{2} T^{8} + 70 p^{3} T^{9} + 40 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - T + 3 p T^{2} + 29 T^{3} + 2123 T^{4} + 4406 T^{5} + 48270 T^{6} + 4406 p T^{7} + 2123 p^{2} T^{8} + 29 p^{3} T^{9} + 3 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 31 T + 553 T^{2} + 235 p T^{3} + 63999 T^{4} + 474070 T^{5} + 2837054 T^{6} + 474070 p T^{7} + 63999 p^{2} T^{8} + 235 p^{4} T^{9} + 553 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 100 T^{2} + 338 T^{3} + 5477 T^{4} + 17234 T^{5} + 210111 T^{6} + 17234 p T^{7} + 5477 p^{2} T^{8} + 338 p^{3} T^{9} + 100 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 22 T + 281 T^{2} + 2288 T^{3} + 13168 T^{4} + 55738 T^{5} + 260260 T^{6} + 55738 p T^{7} + 13168 p^{2} T^{8} + 2288 p^{3} T^{9} + 281 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 33 T + 645 T^{2} + 8957 T^{3} + 96287 T^{4} + 830106 T^{5} + 5865606 T^{6} + 830106 p T^{7} + 96287 p^{2} T^{8} + 8957 p^{3} T^{9} + 645 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 182 T^{2} + 624 T^{3} + 15607 T^{4} + 52904 T^{5} + 826891 T^{6} + 52904 p T^{7} + 15607 p^{2} T^{8} + 624 p^{3} T^{9} + 182 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + 5007 T^{4} - 26388 T^{5} + 325404 T^{6} - 26388 p T^{7} + 5007 p^{2} T^{8} - 6 p^{4} T^{9} + 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 14 T + 154 T^{2} + 654 T^{3} + 823 T^{4} - 55684 T^{5} - 465780 T^{6} - 55684 p T^{7} + 823 p^{2} T^{8} + 654 p^{3} T^{9} + 154 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 8 T + 285 T^{2} - 1760 T^{3} + 36611 T^{4} - 179960 T^{5} + 2743806 T^{6} - 179960 p T^{7} + 36611 p^{2} T^{8} - 1760 p^{3} T^{9} + 285 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 34 T + 772 T^{2} - 12098 T^{3} + 152713 T^{4} - 1545558 T^{5} + 13267723 T^{6} - 1545558 p T^{7} + 152713 p^{2} T^{8} - 12098 p^{3} T^{9} + 772 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 2 T + 292 T^{2} - 130 T^{3} + 37645 T^{4} + 17358 T^{5} + 3024439 T^{6} + 17358 p T^{7} + 37645 p^{2} T^{8} - 130 p^{3} T^{9} + 292 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T + 201 T^{2} - 1225 T^{3} + 17495 T^{4} - 189438 T^{5} + 1160814 T^{6} - 189438 p T^{7} + 17495 p^{2} T^{8} - 1225 p^{3} T^{9} + 201 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 36 T + 869 T^{2} + 14736 T^{3} + 202708 T^{4} + 2250494 T^{5} + 21131980 T^{6} + 2250494 p T^{7} + 202708 p^{2} T^{8} + 14736 p^{3} T^{9} + 869 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 25 T + 624 T^{2} - 120 p T^{3} + 137585 T^{4} - 1471760 T^{5} + 14938465 T^{6} - 1471760 p T^{7} + 137585 p^{2} T^{8} - 120 p^{4} T^{9} + 624 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T + 369 T^{2} - 3356 T^{3} + 61283 T^{4} - 441504 T^{5} + 6208854 T^{6} - 441504 p T^{7} + 61283 p^{2} T^{8} - 3356 p^{3} T^{9} + 369 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 18 T + 315 T^{2} + 3670 T^{3} + 38351 T^{4} + 311400 T^{5} + 3203706 T^{6} + 311400 p T^{7} + 38351 p^{2} T^{8} + 3670 p^{3} T^{9} + 315 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 272 T^{2} - 1810 T^{3} + 40125 T^{4} - 194182 T^{5} + 4458809 T^{6} - 194182 p T^{7} + 40125 p^{2} T^{8} - 1810 p^{3} T^{9} + 272 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.88111418810143856883269873136, −4.15158134724945685176969229163, −4.10714625183307870173084648685, −4.07926617486049631385598195379, −3.94845741008603711473810886192, −3.88246931219256550190283301525, −3.64699729042865675371845661154, −3.55009904622833214281333107844, −3.48098806245053636920383605790, −3.44032428996894061837018443705, −3.37718978040075934511514560743, −3.12837806423087101046958408129, −2.81433713474807584993746053232, −2.65358923007657401724150305089, −2.64729576554148690564309377035, −2.38023927503266082559244423650, −2.23435374553310715129851002979, −1.96135565411831140095907671001, −1.85806453357728590777851309921, −1.80570523006317185469018138457, −1.73354806740713709642732409097, −1.46709187175352079987102822950, −1.29056309510309782633498374646, −1.13960354154361994861709154667, −0.946874585325702323584992885464, 0, 0, 0, 0, 0, 0, 0.946874585325702323584992885464, 1.13960354154361994861709154667, 1.29056309510309782633498374646, 1.46709187175352079987102822950, 1.73354806740713709642732409097, 1.80570523006317185469018138457, 1.85806453357728590777851309921, 1.96135565411831140095907671001, 2.23435374553310715129851002979, 2.38023927503266082559244423650, 2.64729576554148690564309377035, 2.65358923007657401724150305089, 2.81433713474807584993746053232, 3.12837806423087101046958408129, 3.37718978040075934511514560743, 3.44032428996894061837018443705, 3.48098806245053636920383605790, 3.55009904622833214281333107844, 3.64699729042865675371845661154, 3.88246931219256550190283301525, 3.94845741008603711473810886192, 4.07926617486049631385598195379, 4.10714625183307870173084648685, 4.15158134724945685176969229163, 4.88111418810143856883269873136

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

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