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Properties

Label 12-2736e6-1.1-c1e6-0-4
Degree $12$
Conductor $4.195\times 10^{20}$
Sign $1$
Analytic cond. $1.08732\times 10^{8}$
Root an. cond. $4.67408$
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  + 5-s + 4·7-s + 8·11-s + 13-s + 11·17-s − 23-s + 11·25-s − 3·29-s + 12·31-s + 4·35-s − 24·37-s − 19·41-s + 5·43-s − 17·47-s − 2·49-s − 5·53-s + 8·55-s − 13·59-s + 3·61-s + 65-s − 9·67-s + 3·71-s + 11·73-s + 32·77-s − 19·79-s + 24·83-s + 11·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 2.41·11-s + 0.277·13-s + 2.66·17-s − 0.208·23-s + 11/5·25-s − 0.557·29-s + 2.15·31-s + 0.676·35-s − 3.94·37-s − 2.96·41-s + 0.762·43-s − 2.47·47-s − 2/7·49-s − 0.686·53-s + 1.07·55-s − 1.69·59-s + 0.384·61-s + 0.124·65-s − 1.09·67-s + 0.356·71-s + 1.28·73-s + 3.64·77-s − 2.13·79-s + 2.63·83-s + 1.19·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.08732\times 10^{8}\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.44672938\)
\(L(\frac12)\) \(\approx\) \(12.44672938\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + 18 T^{2} + 16 T^{3} + 18 p T^{4} + p^{3} T^{6} \)
good5 \( 1 - T - 2 p T^{2} + p T^{3} + 58 T^{4} - 3 T^{5} - 316 T^{6} - 3 p T^{7} + 58 p^{2} T^{8} + p^{4} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 - 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 4 T + 34 T^{2} - 84 T^{3} + 34 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - T - 6 T^{2} - 107 T^{3} + 8 T^{4} + 363 T^{5} + 6088 T^{6} + 363 p T^{7} + 8 p^{2} T^{8} - 107 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 11 T + 46 T^{2} - 141 T^{3} + 588 T^{4} - 779 T^{5} - 3804 T^{6} - 779 p T^{7} + 588 p^{2} T^{8} - 141 p^{3} T^{9} + 46 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + T - 64 T^{2} - p T^{3} + 2686 T^{4} + 399 T^{5} - 71434 T^{6} + 399 p T^{7} + 2686 p^{2} T^{8} - p^{4} T^{9} - 64 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T - 38 T^{2} - 211 T^{3} + 318 T^{4} + 2617 T^{5} + 8124 T^{6} + 2617 p T^{7} + 318 p^{2} T^{8} - 211 p^{3} T^{9} - 38 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 6 T + 39 T^{2} - 156 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 12 T + 93 T^{2} + 596 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 19 T + 147 T^{2} + 868 T^{3} + 6641 T^{4} + 993 p T^{5} + 5126 p T^{6} + 993 p^{2} T^{7} + 6641 p^{2} T^{8} + 868 p^{3} T^{9} + 147 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 5 T - 20 T^{2} - 53 T^{3} - 340 T^{4} + 145 p T^{5} + 770 p T^{6} + 145 p^{2} T^{7} - 340 p^{2} T^{8} - 53 p^{3} T^{9} - 20 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 17 T + 148 T^{2} + 645 T^{3} - 1638 T^{4} - 1367 p T^{5} - 597786 T^{6} - 1367 p^{2} T^{7} - 1638 p^{2} T^{8} + 645 p^{3} T^{9} + 148 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 5 T - 126 T^{2} - 217 T^{3} + 12368 T^{4} + 7401 T^{5} - 744728 T^{6} + 7401 p T^{7} + 12368 p^{2} T^{8} - 217 p^{3} T^{9} - 126 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 13 T - 57 T^{2} - 4 p T^{3} + 17111 T^{4} + 57231 T^{5} - 672446 T^{6} + 57231 p T^{7} + 17111 p^{2} T^{8} - 4 p^{4} T^{9} - 57 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 74 T^{2} + 967 T^{3} + 2 T^{4} - 31249 T^{5} + 369908 T^{6} - 31249 p T^{7} + 2 p^{2} T^{8} + 967 p^{3} T^{9} - 74 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T - 89 T^{2} - 836 T^{3} + 6863 T^{4} + 36491 T^{5} - 298894 T^{6} + 36491 p T^{7} + 6863 p^{2} T^{8} - 836 p^{3} T^{9} - 89 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T - 200 T^{2} + 217 T^{3} + 27576 T^{4} - 14287 T^{5} - 2252814 T^{6} - 14287 p T^{7} + 27576 p^{2} T^{8} + 217 p^{3} T^{9} - 200 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 11 T - 121 T^{2} + 512 T^{3} + 23441 T^{4} - 54021 T^{5} - 1623378 T^{6} - 54021 p T^{7} + 23441 p^{2} T^{8} + 512 p^{3} T^{9} - 121 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 19 T + 36 T^{2} + 203 T^{3} + 23216 T^{4} + 113379 T^{5} - 695102 T^{6} + 113379 p T^{7} + 23216 p^{2} T^{8} + 203 p^{3} T^{9} + 36 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 12 T + 206 T^{2} - 1360 T^{3} + 206 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 3 T - 74 T^{2} - 269 T^{3} - 468 T^{4} + 21389 T^{5} + 725628 T^{6} + 21389 p T^{7} - 468 p^{2} T^{8} - 269 p^{3} T^{9} - 74 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + T - 273 T^{2} - 112 T^{3} + 48305 T^{4} + 9135 T^{5} - 5429186 T^{6} + 9135 p T^{7} + 48305 p^{2} T^{8} - 112 p^{3} T^{9} - 273 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.75615735146348148548041409403, −4.25338431021862868195134158749, −4.22536350273921261510173141294, −4.11126731528939955740894762806, −4.10212037664701258248807392467, −3.86433503015979073256997699848, −3.55835698384995504430722405664, −3.46456621445416204803558983669, −3.41099635813918224858547667214, −3.28036609212485406099793606371, −3.06192224738585658876255105834, −2.95696627051738520714635018284, −2.82609838140570486517082684665, −2.65291303451834435554026003408, −2.36435130033824793497555709406, −1.98027876713236566087506748135, −1.90041403357328413741456561714, −1.61646414625101143280528728787, −1.57506651254605296634772252973, −1.43826425333825599818937882611, −1.43782517515046950310239998352, −1.15666693086010835671276383774, −0.962741870965517403849428894246, −0.48055133231207046451476981210, −0.32750099656648494969544384122, 0.32750099656648494969544384122, 0.48055133231207046451476981210, 0.962741870965517403849428894246, 1.15666693086010835671276383774, 1.43782517515046950310239998352, 1.43826425333825599818937882611, 1.57506651254605296634772252973, 1.61646414625101143280528728787, 1.90041403357328413741456561714, 1.98027876713236566087506748135, 2.36435130033824793497555709406, 2.65291303451834435554026003408, 2.82609838140570486517082684665, 2.95696627051738520714635018284, 3.06192224738585658876255105834, 3.28036609212485406099793606371, 3.41099635813918224858547667214, 3.46456621445416204803558983669, 3.55835698384995504430722405664, 3.86433503015979073256997699848, 4.10212037664701258248807392467, 4.11126731528939955740894762806, 4.22536350273921261510173141294, 4.25338431021862868195134158749, 4.75615735146348148548041409403

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

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