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Properties

Label 12-3549e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.998\times 10^{21}$
Sign $1$
Analytic cond. $5.17962\times 10^{8}$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 6·3-s − 5·4-s + 3·5-s + 12·6-s − 6·7-s + 14·8-s + 21·9-s − 6·10-s + 30·12-s + 12·14-s − 18·15-s + 7·16-s − 9·17-s − 42·18-s + 11·19-s − 15·20-s + 36·21-s − 11·23-s − 84·24-s + 4·25-s − 56·27-s + 30·28-s − 10·29-s + 36·30-s + 7·31-s − 42·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 3.46·3-s − 5/2·4-s + 1.34·5-s + 4.89·6-s − 2.26·7-s + 4.94·8-s + 7·9-s − 1.89·10-s + 8.66·12-s + 3.20·14-s − 4.64·15-s + 7/4·16-s − 2.18·17-s − 9.89·18-s + 2.52·19-s − 3.35·20-s + 7.85·21-s − 2.29·23-s − 17.1·24-s + 4/5·25-s − 10.7·27-s + 5.66·28-s − 1.85·29-s + 6.57·30-s + 1.25·31-s − 7.42·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(5.17962\times 10^{8}\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
7 \( ( 1 + T )^{6} \)
13 \( 1 \)
good2 \( ( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - 3 T + p T^{2} - 4 p T^{3} + 99 T^{4} - 177 T^{5} + 262 T^{6} - 177 p T^{7} + 99 p^{2} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 19 T^{2} + 42 T^{3} + 173 T^{4} + 952 T^{5} + 1589 T^{6} + 952 p T^{7} + 173 p^{2} T^{8} + 42 p^{3} T^{9} + 19 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 9 T + 4 p T^{2} + 314 T^{3} + 1137 T^{4} + 2817 T^{5} + 9172 T^{6} + 2817 p T^{7} + 1137 p^{2} T^{8} + 314 p^{3} T^{9} + 4 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 120 T^{2} - 814 T^{3} + 5387 T^{4} - 26799 T^{5} + 131848 T^{6} - 26799 p T^{7} + 5387 p^{2} T^{8} - 814 p^{3} T^{9} + 120 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 11 T + 137 T^{2} + 1006 T^{3} + 7520 T^{4} + 41758 T^{5} + 227661 T^{6} + 41758 p T^{7} + 7520 p^{2} T^{8} + 1006 p^{3} T^{9} + 137 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 10 T + 107 T^{2} + 716 T^{3} + 5165 T^{4} + 30938 T^{5} + 181023 T^{6} + 30938 p T^{7} + 5165 p^{2} T^{8} + 716 p^{3} T^{9} + 107 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 7 T + 90 T^{2} - 420 T^{3} + 2643 T^{4} - 10101 T^{5} + 54940 T^{6} - 10101 p T^{7} + 2643 p^{2} T^{8} - 420 p^{3} T^{9} + 90 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 20 T + 302 T^{2} + 3217 T^{3} + 29167 T^{4} + 218134 T^{5} + 1440055 T^{6} + 218134 p T^{7} + 29167 p^{2} T^{8} + 3217 p^{3} T^{9} + 302 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 10 T + 201 T^{2} + 1619 T^{3} + 18191 T^{4} + 116863 T^{5} + 951886 T^{6} + 116863 p T^{7} + 18191 p^{2} T^{8} + 1619 p^{3} T^{9} + 201 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T + 130 T^{2} + 24 p T^{3} + 11707 T^{4} + 71715 T^{5} + 576724 T^{6} + 71715 p T^{7} + 11707 p^{2} T^{8} + 24 p^{4} T^{9} + 130 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 36 T + 683 T^{2} - 8723 T^{3} + 84469 T^{4} - 675867 T^{5} + 4828054 T^{6} - 675867 p T^{7} + 84469 p^{2} T^{8} - 8723 p^{3} T^{9} + 683 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 240 T^{2} + 2271 T^{3} + 28343 T^{4} + 209118 T^{5} + 1914389 T^{6} + 209118 p T^{7} + 28343 p^{2} T^{8} + 2271 p^{3} T^{9} + 240 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 41 T + 849 T^{2} + 12026 T^{3} + 136723 T^{4} + 1332761 T^{5} + 11131902 T^{6} + 1332761 p T^{7} + 136723 p^{2} T^{8} + 12026 p^{3} T^{9} + 849 p^{4} T^{10} + 41 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 24 T + 459 T^{2} + 5891 T^{3} + 66989 T^{4} + 611329 T^{5} + 5203358 T^{6} + 611329 p T^{7} + 66989 p^{2} T^{8} + 5891 p^{3} T^{9} + 459 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 15 T + 313 T^{2} - 3342 T^{3} + 43558 T^{4} - 362172 T^{5} + 3588451 T^{6} - 362172 p T^{7} + 43558 p^{2} T^{8} - 3342 p^{3} T^{9} + 313 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 13 T + 277 T^{2} - 2194 T^{3} + 27790 T^{4} - 145082 T^{5} + 1885745 T^{6} - 145082 p T^{7} + 27790 p^{2} T^{8} - 2194 p^{3} T^{9} + 277 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 25 T + 8 p T^{2} - 8474 T^{3} + 115993 T^{4} - 1191113 T^{5} + 11546764 T^{6} - 1191113 p T^{7} + 115993 p^{2} T^{8} - 8474 p^{3} T^{9} + 8 p^{5} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 349 T^{2} + 4840 T^{3} + 62917 T^{4} + 661804 T^{5} + 6479605 T^{6} + 661804 p T^{7} + 62917 p^{2} T^{8} + 4840 p^{3} T^{9} + 349 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 2 T + 273 T^{2} + 311 T^{3} + 41977 T^{4} + 31061 T^{5} + 4131570 T^{6} + 31061 p T^{7} + 41977 p^{2} T^{8} + 311 p^{3} T^{9} + 273 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 15 T + 269 T^{2} - 4632 T^{3} + 52315 T^{4} - 594097 T^{5} + 6409446 T^{6} - 594097 p T^{7} + 52315 p^{2} T^{8} - 4632 p^{3} T^{9} + 269 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 10 T + 495 T^{2} - 3761 T^{3} + 106321 T^{4} - 634849 T^{5} + 13148462 T^{6} - 634849 p T^{7} + 106321 p^{2} T^{8} - 3761 p^{3} T^{9} + 495 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.94989930546292531078830454732, −4.75543171884389173681946004104, −4.56708094070636324918409868809, −4.48141750969445470055023812961, −4.45583999999708763625031473584, −4.29948351459634190780210854686, −4.04904185215980668875521215313, −3.78059969023679556829698775415, −3.64308388771823921610779274164, −3.61641658740644352148903107881, −3.60506058322161640535281476863, −3.58135840048315661923033492461, −3.16264613408528228451104245586, −3.04803274501233396343879680677, −2.61851235544014838978709404119, −2.51832252966962291920432052055, −2.23039603214137937582811471081, −2.20211483159811529039092401757, −2.09107497882958714991519526211, −1.71267112218039215527183958272, −1.40187526880167692192173435658, −1.21471146465271900786027555825, −1.21116286435292637344351639245, −1.02553360461117411126271039035, −0.903689445602711119980598654167, 0, 0, 0, 0, 0, 0, 0.903689445602711119980598654167, 1.02553360461117411126271039035, 1.21116286435292637344351639245, 1.21471146465271900786027555825, 1.40187526880167692192173435658, 1.71267112218039215527183958272, 2.09107497882958714991519526211, 2.20211483159811529039092401757, 2.23039603214137937582811471081, 2.51832252966962291920432052055, 2.61851235544014838978709404119, 3.04803274501233396343879680677, 3.16264613408528228451104245586, 3.58135840048315661923033492461, 3.60506058322161640535281476863, 3.61641658740644352148903107881, 3.64308388771823921610779274164, 3.78059969023679556829698775415, 4.04904185215980668875521215313, 4.29948351459634190780210854686, 4.45583999999708763625031473584, 4.48141750969445470055023812961, 4.56708094070636324918409868809, 4.75543171884389173681946004104, 4.94989930546292531078830454732

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

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