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Properties

Label 12-2912e6-1.1-c1e6-0-1
Degree $12$
Conductor $6.097\times 10^{20}$
Sign $1$
Analytic cond. $1.58055\times 10^{8}$
Root an. cond. $4.82207$
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  + 4·3-s + 3·5-s + 6·7-s + 3·9-s + 8·11-s − 6·13-s + 12·15-s − 2·17-s + 9·19-s + 24·21-s − 11·23-s − 6·27-s + 7·29-s + 31-s + 32·33-s + 18·35-s + 16·37-s − 24·39-s + 43-s + 9·45-s − 11·47-s + 21·49-s − 8·51-s + 23·53-s + 24·55-s + 36·57-s + 10·59-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.34·5-s + 2.26·7-s + 9-s + 2.41·11-s − 1.66·13-s + 3.09·15-s − 0.485·17-s + 2.06·19-s + 5.23·21-s − 2.29·23-s − 1.15·27-s + 1.29·29-s + 0.179·31-s + 5.57·33-s + 3.04·35-s + 2.63·37-s − 3.84·39-s + 0.152·43-s + 1.34·45-s − 1.60·47-s + 3·49-s − 1.12·51-s + 3.15·53-s + 3.23·55-s + 4.76·57-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(1.58055\times 10^{8}\)
Root analytic conductor: \(4.82207\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(79.16105756\)
\(L(\frac12)\) \(\approx\) \(79.16105756\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 - T )^{6} \)
13 \( ( 1 + T )^{6} \)
good3 \( 1 - 4 T + 13 T^{2} - 34 T^{3} + 25 p T^{4} - 148 T^{5} + 278 T^{6} - 148 p T^{7} + 25 p^{3} T^{8} - 34 p^{3} T^{9} + 13 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 9 T^{2} - 24 T^{3} + 37 T^{4} - 41 T^{5} + 138 T^{6} - 41 p T^{7} + 37 p^{2} T^{8} - 24 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 53 T^{2} - 216 T^{3} + 873 T^{4} - 2654 T^{5} + 9578 T^{6} - 2654 p T^{7} + 873 p^{2} T^{8} - 216 p^{3} T^{9} + 53 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 2 T^{2} + 52 T^{3} + 435 T^{4} + 802 T^{5} + 1508 T^{6} + 802 p T^{7} + 435 p^{2} T^{8} + 52 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 9 T + 123 T^{2} - 778 T^{3} + 5983 T^{4} - 28265 T^{5} + 152490 T^{6} - 28265 p T^{7} + 5983 p^{2} T^{8} - 778 p^{3} T^{9} + 123 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 11 T + 140 T^{2} + 907 T^{3} + 6616 T^{4} + 31195 T^{5} + 179346 T^{6} + 31195 p T^{7} + 6616 p^{2} T^{8} + 907 p^{3} T^{9} + 140 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 7 T + 79 T^{2} - 460 T^{3} + 3469 T^{4} - 20745 T^{5} + 125902 T^{6} - 20745 p T^{7} + 3469 p^{2} T^{8} - 460 p^{3} T^{9} + 79 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - T + 36 T^{2} - 91 T^{3} + 2032 T^{4} - 3861 T^{5} + 44270 T^{6} - 3861 p T^{7} + 2032 p^{2} T^{8} - 91 p^{3} T^{9} + 36 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 16 T + 207 T^{2} - 2150 T^{3} + 18553 T^{4} - 138836 T^{5} + 901058 T^{6} - 138836 p T^{7} + 18553 p^{2} T^{8} - 2150 p^{3} T^{9} + 207 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 109 T^{2} + 228 T^{3} + 7003 T^{4} + 14876 T^{5} + 348974 T^{6} + 14876 p T^{7} + 7003 p^{2} T^{8} + 228 p^{3} T^{9} + 109 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - T + 101 T^{2} - 362 T^{3} + 6127 T^{4} - 33885 T^{5} + 257846 T^{6} - 33885 p T^{7} + 6127 p^{2} T^{8} - 362 p^{3} T^{9} + 101 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 11 T + 290 T^{2} + 2331 T^{3} + 33762 T^{4} + 206255 T^{5} + 2099750 T^{6} + 206255 p T^{7} + 33762 p^{2} T^{8} + 2331 p^{3} T^{9} + 290 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 23 T + 367 T^{2} - 3524 T^{3} + 27065 T^{4} - 148361 T^{5} + 1001126 T^{6} - 148361 p T^{7} + 27065 p^{2} T^{8} - 3524 p^{3} T^{9} + 367 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 10 T + 206 T^{2} - 1122 T^{3} + 17815 T^{4} - 75760 T^{5} + 1189604 T^{6} - 75760 p T^{7} + 17815 p^{2} T^{8} - 1122 p^{3} T^{9} + 206 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 10 T + 315 T^{2} - 2958 T^{3} + 43899 T^{4} - 354800 T^{5} + 3465522 T^{6} - 354800 p T^{7} + 43899 p^{2} T^{8} - 2958 p^{3} T^{9} + 315 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 8 T + 231 T^{2} + 1626 T^{3} + 29791 T^{4} + 174938 T^{5} + 2370770 T^{6} + 174938 p T^{7} + 29791 p^{2} T^{8} + 1626 p^{3} T^{9} + 231 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 14 T + 360 T^{2} + 3956 T^{3} + 58775 T^{4} + 502302 T^{5} + 5436448 T^{6} + 502302 p T^{7} + 58775 p^{2} T^{8} + 3956 p^{3} T^{9} + 360 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 11 T + 362 T^{2} - 3163 T^{3} + 59400 T^{4} - 412635 T^{5} + 5564696 T^{6} - 412635 p T^{7} + 59400 p^{2} T^{8} - 3163 p^{3} T^{9} + 362 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 3 T + 276 T^{2} + 311 T^{3} + 40416 T^{4} + 32339 T^{5} + 3946882 T^{6} + 32339 p T^{7} + 40416 p^{2} T^{8} + 311 p^{3} T^{9} + 276 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 17 T + 365 T^{2} - 3744 T^{3} + 46039 T^{4} - 346547 T^{5} + 3758486 T^{6} - 346547 p T^{7} + 46039 p^{2} T^{8} - 3744 p^{3} T^{9} + 365 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 5 T + 315 T^{2} + 1834 T^{3} + 54165 T^{4} + 271585 T^{5} + 6030110 T^{6} + 271585 p T^{7} + 54165 p^{2} T^{8} + 1834 p^{3} T^{9} + 315 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 27 T + 442 T^{2} - 5943 T^{3} + 76968 T^{4} - 796947 T^{5} + 7777144 T^{6} - 796947 p T^{7} + 76968 p^{2} T^{8} - 5943 p^{3} T^{9} + 442 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.57348409212262994494091137278, −4.44275338540049671957003652361, −4.20927713010766573780605855384, −3.97694157390528262039652150834, −3.96197383282625020163035862961, −3.79774274754808075225575517831, −3.77176234295749890791693432852, −3.31706981567630209495020157724, −3.22577759679866068691189180729, −3.11384936746279122772940695443, −2.94650372121103550797032734636, −2.89651285683384159092366003419, −2.81755332949796436347385657169, −2.31561362412517742739217597782, −2.20258374025434159864189575895, −2.18745111781705875275234831744, −2.04061188540342989536818986457, −1.98738017849708977890729334533, −1.87374078629013338432247151081, −1.66011703031324062654119157597, −1.17845688060950487638152129662, −1.04914288057087204065348656667, −0.903604724457015517218129655277, −0.61472584476373795240474166662, −0.55885779477119716928627686744, 0.55885779477119716928627686744, 0.61472584476373795240474166662, 0.903604724457015517218129655277, 1.04914288057087204065348656667, 1.17845688060950487638152129662, 1.66011703031324062654119157597, 1.87374078629013338432247151081, 1.98738017849708977890729334533, 2.04061188540342989536818986457, 2.18745111781705875275234831744, 2.20258374025434159864189575895, 2.31561362412517742739217597782, 2.81755332949796436347385657169, 2.89651285683384159092366003419, 2.94650372121103550797032734636, 3.11384936746279122772940695443, 3.22577759679866068691189180729, 3.31706981567630209495020157724, 3.77176234295749890791693432852, 3.79774274754808075225575517831, 3.96197383282625020163035862961, 3.97694157390528262039652150834, 4.20927713010766573780605855384, 4.44275338540049671957003652361, 4.57348409212262994494091137278

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

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