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Properties

Label 12-8025e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.671\times 10^{23}$
Sign $1$
Analytic cond. $6.92361\times 10^{10}$
Root an. cond. $8.00499$
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  − 3·2-s − 6·3-s + 2·4-s + 18·6-s + 3·8-s + 21·9-s + 6·11-s − 12·12-s + 8·13-s − 8·16-s − 4·17-s − 63·18-s − 4·19-s − 18·22-s − 14·23-s − 18·24-s − 24·26-s − 56·27-s + 10·29-s + 12·31-s + 11·32-s − 36·33-s + 12·34-s + 42·36-s + 12·37-s + 12·38-s − 48·39-s + ⋯
L(s)  = 1  − 2.12·2-s − 3.46·3-s + 4-s + 7.34·6-s + 1.06·8-s + 7·9-s + 1.80·11-s − 3.46·12-s + 2.21·13-s − 2·16-s − 0.970·17-s − 14.8·18-s − 0.917·19-s − 3.83·22-s − 2.91·23-s − 3.67·24-s − 4.70·26-s − 10.7·27-s + 1.85·29-s + 2.15·31-s + 1.94·32-s − 6.26·33-s + 2.05·34-s + 7·36-s + 1.97·37-s + 1.94·38-s − 7.68·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 107^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 107^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12} \cdot 107^{6}\)
Sign: $1$
Analytic conductor: \(6.92361\times 10^{10}\)
Root analytic conductor: \(8.00499\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} \cdot 107^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.037046734\)
\(L(\frac12)\) \(\approx\) \(1.037046734\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( 1 \)
107 \( ( 1 - T )^{6} \)
good2 \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + 21 T^{4} + 31 T^{5} + 47 T^{6} + 31 p T^{7} + 21 p^{2} T^{8} + 3 p^{5} T^{9} + 7 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 27 T^{2} - 18 T^{3} + 328 T^{4} - 52 p T^{5} + 2628 T^{6} - 52 p^{2} T^{7} + 328 p^{2} T^{8} - 18 p^{3} T^{9} + 27 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 6 T + 45 T^{2} - 192 T^{3} + 884 T^{4} - 3338 T^{5} + 11856 T^{6} - 3338 p T^{7} + 884 p^{2} T^{8} - 192 p^{3} T^{9} + 45 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 74 T^{2} - 398 T^{3} + 2307 T^{4} - 9262 T^{5} + 39005 T^{6} - 9262 p T^{7} + 2307 p^{2} T^{8} - 398 p^{3} T^{9} + 74 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 52 T^{2} + 304 T^{3} + 1653 T^{4} + 502 p T^{5} + 36449 T^{6} + 502 p^{2} T^{7} + 1653 p^{2} T^{8} + 304 p^{3} T^{9} + 52 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T + 97 T^{2} + 352 T^{3} + 4159 T^{4} + 12860 T^{5} + 101710 T^{6} + 12860 p T^{7} + 4159 p^{2} T^{8} + 352 p^{3} T^{9} + 97 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 14 T + 169 T^{2} + 1344 T^{3} + 9816 T^{4} + 56322 T^{5} + 300332 T^{6} + 56322 p T^{7} + 9816 p^{2} T^{8} + 1344 p^{3} T^{9} + 169 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 10 T + 106 T^{2} - 666 T^{3} + 5127 T^{4} - 31284 T^{5} + 196460 T^{6} - 31284 p T^{7} + 5127 p^{2} T^{8} - 666 p^{3} T^{9} + 106 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 12 T + 83 T^{2} - 234 T^{3} + 404 T^{4} - 3002 T^{5} + 31572 T^{6} - 3002 p T^{7} + 404 p^{2} T^{8} - 234 p^{3} T^{9} + 83 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 12 T + 178 T^{2} - 1450 T^{3} + 13459 T^{4} - 87006 T^{5} + 620629 T^{6} - 87006 p T^{7} + 13459 p^{2} T^{8} - 1450 p^{3} T^{9} + 178 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 170 T^{2} + 526 T^{3} + 10767 T^{4} + 12092 T^{5} + 448428 T^{6} + 12092 p T^{7} + 10767 p^{2} T^{8} + 526 p^{3} T^{9} + 170 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 12 T + 155 T^{2} - 986 T^{3} + 7828 T^{4} - 39914 T^{5} + 317900 T^{6} - 39914 p T^{7} + 7828 p^{2} T^{8} - 986 p^{3} T^{9} + 155 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 241 T^{2} + 2136 T^{3} + 19920 T^{4} + 133866 T^{5} + 1042664 T^{6} + 133866 p T^{7} + 19920 p^{2} T^{8} + 2136 p^{3} T^{9} + 241 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 274 T^{2} + 2572 T^{3} + 32231 T^{4} + 242456 T^{5} + 2177212 T^{6} + 242456 p T^{7} + 32231 p^{2} T^{8} + 2572 p^{3} T^{9} + 274 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 8 T + 202 T^{2} - 1944 T^{3} + 24615 T^{4} - 184624 T^{5} + 1909772 T^{6} - 184624 p T^{7} + 24615 p^{2} T^{8} - 1944 p^{3} T^{9} + 202 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 24 T + 234 T^{2} + 614 T^{3} - 3901 T^{4} - 4650 T^{5} + 291029 T^{6} - 4650 p T^{7} - 3901 p^{2} T^{8} + 614 p^{3} T^{9} + 234 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 4 T + 190 T^{2} + 52 T^{3} + 19175 T^{4} + 13448 T^{5} + 1629764 T^{6} + 13448 p T^{7} + 19175 p^{2} T^{8} + 52 p^{3} T^{9} + 190 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 36 T + 701 T^{2} - 9448 T^{3} + 100523 T^{4} - 917924 T^{5} + 7863710 T^{6} - 917924 p T^{7} + 100523 p^{2} T^{8} - 9448 p^{3} T^{9} + 701 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 26 T + 474 T^{2} - 6466 T^{3} + 78623 T^{4} - 792436 T^{5} + 7258924 T^{6} - 792436 p T^{7} + 78623 p^{2} T^{8} - 6466 p^{3} T^{9} + 474 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 8 T + 346 T^{2} - 2968 T^{3} + 56559 T^{4} - 446608 T^{5} + 5607724 T^{6} - 446608 p T^{7} + 56559 p^{2} T^{8} - 2968 p^{3} T^{9} + 346 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 8 T + 297 T^{2} - 1748 T^{3} + 40320 T^{4} - 187722 T^{5} + 3767296 T^{6} - 187722 p T^{7} + 40320 p^{2} T^{8} - 1748 p^{3} T^{9} + 297 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 8 T + 294 T^{2} + 1256 T^{3} + 35167 T^{4} + 53328 T^{5} + 2975252 T^{6} + 53328 p T^{7} + 35167 p^{2} T^{8} + 1256 p^{3} T^{9} + 294 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 24 T + 674 T^{2} - 10296 T^{3} + 164351 T^{4} - 1832752 T^{5} + 21000252 T^{6} - 1832752 p T^{7} + 164351 p^{2} T^{8} - 10296 p^{3} T^{9} + 674 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.18123800765002896860446782073, −3.93196848521818382439071364465, −3.72305147536855569343835741444, −3.60133864035887571136247411515, −3.54673565066212977020954508294, −3.49522918779423517167268884585, −3.16770383488663213719903895297, −3.14350044432454804815365707633, −2.76688233759201853894712681931, −2.72445624445941355850399006519, −2.47635715730191173769336269370, −2.28133375852782909493264761494, −2.14836915052741723440114482211, −1.91271817198778481521345357915, −1.81256537425829669552106446697, −1.50537385092243113753026418619, −1.48499574397044992406608499994, −1.44442978736190380143411822721, −1.39335634367080691678044314597, −0.992044254788236396057348471244, −0.61342083677087937045406512310, −0.50566266884849300045712985438, −0.50106048587793947885371909086, −0.49722238026425814567641926919, −0.43749603232022218979813244324, 0.43749603232022218979813244324, 0.49722238026425814567641926919, 0.50106048587793947885371909086, 0.50566266884849300045712985438, 0.61342083677087937045406512310, 0.992044254788236396057348471244, 1.39335634367080691678044314597, 1.44442978736190380143411822721, 1.48499574397044992406608499994, 1.50537385092243113753026418619, 1.81256537425829669552106446697, 1.91271817198778481521345357915, 2.14836915052741723440114482211, 2.28133375852782909493264761494, 2.47635715730191173769336269370, 2.72445624445941355850399006519, 2.76688233759201853894712681931, 3.14350044432454804815365707633, 3.16770383488663213719903895297, 3.49522918779423517167268884585, 3.54673565066212977020954508294, 3.60133864035887571136247411515, 3.72305147536855569343835741444, 3.93196848521818382439071364465, 4.18123800765002896860446782073

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

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