Properties

Label 12-507e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.698\times 10^{16}$
Sign $1$
Analytic cond. $4402.61$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·3-s + 2·4-s − 12·5-s + 9·6-s + 2·7-s − 7·8-s + 3·9-s − 36·10-s + 5·11-s + 6·12-s + 6·14-s − 36·15-s − 14·16-s + 17-s + 9·18-s − 7·19-s − 24·20-s + 6·21-s + 15·22-s − 21·24-s + 68·25-s − 2·27-s + 4·28-s + 2·29-s − 108·30-s + 32·31-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 4-s − 5.36·5-s + 3.67·6-s + 0.755·7-s − 2.47·8-s + 9-s − 11.3·10-s + 1.50·11-s + 1.73·12-s + 1.60·14-s − 9.29·15-s − 7/2·16-s + 0.242·17-s + 2.12·18-s − 1.60·19-s − 5.36·20-s + 1.30·21-s + 3.19·22-s − 4.28·24-s + 68/5·25-s − 0.384·27-s + 0.755·28-s + 0.371·29-s − 19.7·30-s + 5.74·31-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4402.61\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.118443644\)
\(L(\frac12)\) \(\approx\) \(4.118443644\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 \)
good2 \( 1 - 3 T + 7 T^{2} - p^{3} T^{3} + 3 T^{4} + 7 p T^{5} - 27 T^{6} + 7 p^{2} T^{7} + 3 p^{2} T^{8} - p^{6} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + 6 T + 4 p T^{2} + 47 T^{3} + 4 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 2 T - 16 T^{2} + 2 p T^{3} + 206 T^{4} - 78 T^{5} - 1581 T^{6} - 78 p T^{7} + 206 p^{2} T^{8} + 2 p^{4} T^{9} - 16 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 5 T + 13 T^{3} + 5 T^{4} + 420 T^{5} - 2477 T^{6} + 420 p T^{7} + 5 p^{2} T^{8} + 13 p^{3} T^{9} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - T - 2 p T^{2} + 59 T^{3} + 583 T^{4} - 744 T^{5} - 9079 T^{6} - 744 p T^{7} + 583 p^{2} T^{8} + 59 p^{3} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 7 T - 22 T^{2} - 49 T^{3} + 1781 T^{4} + 3024 T^{5} - 25221 T^{6} + 3024 p T^{7} + 1781 p^{2} T^{8} - 49 p^{3} T^{9} - 22 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 20 T^{2} + 182 T^{3} - 60 T^{4} - 1820 T^{5} + 23415 T^{6} - 1820 p T^{7} - 60 p^{2} T^{8} + 182 p^{3} T^{9} - 20 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 2 T - 68 T^{2} + 30 T^{3} + 2922 T^{4} + 14 p T^{5} - 3357 p T^{6} + 14 p^{2} T^{7} + 2922 p^{2} T^{8} + 30 p^{3} T^{9} - 68 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 16 T + 134 T^{2} - 795 T^{3} + 134 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 22 T + 214 T^{2} - 1930 T^{3} + 18800 T^{4} - 131908 T^{5} + 761435 T^{6} - 131908 p T^{7} + 18800 p^{2} T^{8} - 1930 p^{3} T^{9} + 214 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 11 T - 26 T^{2} + 129 T^{3} + 5979 T^{4} - 14228 T^{5} - 189399 T^{6} - 14228 p T^{7} + 5979 p^{2} T^{8} + 129 p^{3} T^{9} - 26 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 15 T + 49 T^{2} + 22 T^{3} + 3443 T^{4} - 34951 T^{5} + 182582 T^{6} - 34951 p T^{7} + 3443 p^{2} T^{8} + 22 p^{3} T^{9} + 49 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 7 T + 155 T^{2} + 665 T^{3} + 155 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 6 T - 125 T^{2} - 242 T^{3} + 12798 T^{4} + 4142 T^{5} - 879081 T^{6} + 4142 p T^{7} + 12798 p^{2} T^{8} - 242 p^{3} T^{9} - 125 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 13 T + 2 T^{2} + 667 T^{3} - 745 T^{4} - 20484 T^{5} + 137445 T^{6} - 20484 p T^{7} - 745 p^{2} T^{8} + 667 p^{3} T^{9} + 2 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 11 T - 34 T^{2} + 1325 T^{3} - 3025 T^{4} - 55734 T^{5} + 743907 T^{6} - 55734 p T^{7} - 3025 p^{2} T^{8} + 1325 p^{3} T^{9} - 34 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 122 T^{2} - 406 T^{3} + 6222 T^{4} + 24766 T^{5} - 299733 T^{6} + 24766 p T^{7} + 6222 p^{2} T^{8} - 406 p^{3} T^{9} - 122 p^{4} T^{10} + p^{6} T^{12} \)
73 \( ( 1 + 6 T + 84 T^{2} - 47 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 12 T + 290 T^{2} + 2035 T^{3} + 290 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - T - 166 T^{2} - 37 T^{3} + 12961 T^{4} + 10950 T^{5} - 1075879 T^{6} + 10950 p T^{7} + 12961 p^{2} T^{8} - 37 p^{3} T^{9} - 166 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 5 T + 15 T^{2} + 1384 T^{3} + 2465 T^{4} + 30675 T^{5} + 2025310 T^{6} + 30675 p T^{7} + 2465 p^{2} T^{8} + 1384 p^{3} T^{9} + 15 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.82622542458376858618406577847, −5.73266012847901523304275944795, −5.68755344814861957342415173623, −5.00866379070956067825693471656, −4.88694565411790682803627033180, −4.77135688358494427192761258073, −4.43359241058474516446512670875, −4.31191163403381028596272759754, −4.28106571445404714140991828176, −4.24004163345415151061307309970, −4.15746545763124965180406677480, −4.12798171446516750194767231604, −3.93102796248669076083918032391, −3.36801796273384047003738867504, −3.33581714242086416098020702785, −3.28599081964485661315101016379, −2.91388535327318393574000253626, −2.91006810726655084707858498355, −2.81355308710825158780751388134, −2.34177057312061585192384103210, −2.27421361911415527655537154334, −1.32037328581597356234840187406, −1.18866325473658283854390771567, −0.56154658524569404100272218791, −0.54441248363324078977583531786, 0.54441248363324078977583531786, 0.56154658524569404100272218791, 1.18866325473658283854390771567, 1.32037328581597356234840187406, 2.27421361911415527655537154334, 2.34177057312061585192384103210, 2.81355308710825158780751388134, 2.91006810726655084707858498355, 2.91388535327318393574000253626, 3.28599081964485661315101016379, 3.33581714242086416098020702785, 3.36801796273384047003738867504, 3.93102796248669076083918032391, 4.12798171446516750194767231604, 4.15746545763124965180406677480, 4.24004163345415151061307309970, 4.28106571445404714140991828176, 4.31191163403381028596272759754, 4.43359241058474516446512670875, 4.77135688358494427192761258073, 4.88694565411790682803627033180, 5.00866379070956067825693471656, 5.68755344814861957342415173623, 5.73266012847901523304275944795, 5.82622542458376858618406577847

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

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