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Properties

Label 12-273e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.140\times 10^{14}$
Sign $1$
Analytic cond. $107.309$
Root an. cond. $1.47645$
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  + 3·3-s + 4-s + 4·5-s − 3·7-s − 2·8-s + 3·9-s − 8·11-s + 3·12-s + 10·13-s + 12·15-s + 2·16-s − 12·17-s − 3·19-s + 4·20-s − 9·21-s + 7·23-s − 6·24-s − 2·27-s − 3·28-s − 9·29-s + 10·31-s − 3·32-s − 24·33-s − 12·35-s + 3·36-s + 30·39-s − 8·40-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/2·4-s + 1.78·5-s − 1.13·7-s − 0.707·8-s + 9-s − 2.41·11-s + 0.866·12-s + 2.77·13-s + 3.09·15-s + 1/2·16-s − 2.91·17-s − 0.688·19-s + 0.894·20-s − 1.96·21-s + 1.45·23-s − 1.22·24-s − 0.384·27-s − 0.566·28-s − 1.67·29-s + 1.79·31-s − 0.530·32-s − 4.17·33-s − 2.02·35-s + 1/2·36-s + 4.80·39-s − 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(107.309\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.605279155\)
\(L(\frac12)\) \(\approx\) \(2.605279155\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{3} \)
7 \( ( 1 + T + T^{2} )^{3} \)
13 \( 1 - 10 T + 62 T^{2} - 265 T^{3} + 62 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - T^{2} + p T^{3} - T^{4} - T^{5} + 15 T^{6} - p T^{7} - p^{2} T^{8} + p^{4} T^{9} - p^{4} T^{10} + p^{6} T^{12} \)
5 \( ( 1 - 2 T + 6 T^{2} - 11 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( 1 + 8 T + 20 T^{2} + 2 p T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + 7 p^{2} T^{6} + 4 p^{3} T^{7} + 12 p^{3} T^{8} + 2 p^{4} T^{9} + 20 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 12 T + 50 T^{2} + 226 T^{3} + 1826 T^{4} + 7510 T^{5} + 21399 T^{6} + 7510 p T^{7} + 1826 p^{2} T^{8} + 226 p^{3} T^{9} + 50 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 31 T^{2} - 54 T^{3} + 579 T^{4} + 3 T^{5} - 12362 T^{6} + 3 p T^{7} + 579 p^{2} T^{8} - 54 p^{3} T^{9} - 31 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 7 T - 28 T^{2} + 75 T^{3} + 2009 T^{4} - 2548 T^{5} - 43705 T^{6} - 2548 p T^{7} + 2009 p^{2} T^{8} + 75 p^{3} T^{9} - 28 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T - 28 T^{2} - 89 T^{3} + 3905 T^{4} + 8986 T^{5} - 77667 T^{6} + 8986 p T^{7} + 3905 p^{2} T^{8} - 89 p^{3} T^{9} - 28 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 5 T + 57 T^{2} - 141 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 22 T^{2} + 254 T^{3} - 330 T^{4} - 2794 T^{5} + 99527 T^{6} - 2794 p T^{7} - 330 p^{2} T^{8} + 254 p^{3} T^{9} - 22 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 6 T - 19 T^{2} - 222 T^{3} - 886 T^{4} - 1794 T^{5} + 29477 T^{6} - 1794 p T^{7} - 886 p^{2} T^{8} - 222 p^{3} T^{9} - 19 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 7 T - 88 T^{2} + 5 p T^{3} + 8769 T^{4} - 10888 T^{5} - 394085 T^{6} - 10888 p T^{7} + 8769 p^{2} T^{8} + 5 p^{4} T^{9} - 88 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 9 T + 97 T^{2} - 441 T^{3} + 97 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 13 T + 182 T^{2} + 1381 T^{3} + 182 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 11 T + 102 T^{2} + 883 T^{3} + 2947 T^{4} + 7226 T^{5} + 81827 T^{6} + 7226 p T^{7} + 2947 p^{2} T^{8} + 883 p^{3} T^{9} + 102 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 19 T + 123 T^{2} + 436 T^{3} + 3309 T^{4} + 5249 T^{5} - 170754 T^{6} + 5249 p T^{7} + 3309 p^{2} T^{8} + 436 p^{3} T^{9} + 123 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 21 T + 138 T^{2} + 733 T^{3} + 12393 T^{4} + 94572 T^{5} + 394899 T^{6} + 94572 p T^{7} + 12393 p^{2} T^{8} + 733 p^{3} T^{9} + 138 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 22 T + 144 T^{2} + 794 T^{3} + 17914 T^{4} + 185362 T^{5} + 1244027 T^{6} + 185362 p T^{7} + 17914 p^{2} T^{8} + 794 p^{3} T^{9} + 144 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 14 T + 276 T^{2} + 2115 T^{3} + 276 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + T + 175 T^{2} + 321 T^{3} + 175 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 32 T + 528 T^{2} - 5677 T^{3} + 528 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 3 T - 181 T^{2} + 212 T^{3} + 18017 T^{4} + 3407 T^{5} - 1721370 T^{6} + 3407 p T^{7} + 18017 p^{2} T^{8} + 212 p^{3} T^{9} - 181 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 21 T + 96 T^{2} + 157 T^{3} + 7839 T^{4} - 68904 T^{5} - 172047 T^{6} - 68904 p T^{7} + 7839 p^{2} T^{8} + 157 p^{3} T^{9} + 96 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.33715071824402204123908696742, −6.28361021389968667566908547506, −6.15276237210248222872376886924, −6.04077659450855728669164355594, −6.00572879362743356790748196060, −5.89425244752649667088649730763, −5.55350360634906955875292657616, −5.02169251533633956181670026240, −4.91369729691846214089469724401, −4.87560986069220489586348275210, −4.71928688702739114886362319259, −4.20251468133369929335861778013, −4.04149677556930015792589105553, −3.92633861587325962106111808087, −3.57169598762409832541743699862, −3.18939263938219306776796529479, −2.98410815948756040506829103718, −2.90397292619631800623651374042, −2.89755575154883332849319403192, −2.65582446043950492361987166498, −2.06283378374686996872715104042, −1.90052012833032333917315447055, −1.82138220819548978445135977829, −1.52486228965156254413938250320, −0.42576099935636519415210507762, 0.42576099935636519415210507762, 1.52486228965156254413938250320, 1.82138220819548978445135977829, 1.90052012833032333917315447055, 2.06283378374686996872715104042, 2.65582446043950492361987166498, 2.89755575154883332849319403192, 2.90397292619631800623651374042, 2.98410815948756040506829103718, 3.18939263938219306776796529479, 3.57169598762409832541743699862, 3.92633861587325962106111808087, 4.04149677556930015792589105553, 4.20251468133369929335861778013, 4.71928688702739114886362319259, 4.87560986069220489586348275210, 4.91369729691846214089469724401, 5.02169251533633956181670026240, 5.55350360634906955875292657616, 5.89425244752649667088649730763, 6.00572879362743356790748196060, 6.04077659450855728669164355594, 6.15276237210248222872376886924, 6.28361021389968667566908547506, 6.33715071824402204123908696742

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

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