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Properties

Label 12-819e6-1.1-c1e6-0-7
Degree $12$
Conductor $3.018\times 10^{17}$
Sign $1$
Analytic cond. $78228.6$
Root an. cond. $2.55729$
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  + 3·4-s + 3·5-s − 2·8-s − 3·11-s + 6·13-s + 6·16-s + 9·17-s − 3·19-s + 9·20-s + 15·25-s − 12·29-s − 12·31-s − 9·32-s + 3·37-s − 6·40-s − 12·43-s − 9·44-s + 6·47-s + 18·52-s − 3·53-s − 9·55-s − 3·59-s − 15·61-s + 14·64-s + 18·65-s + 9·67-s + 27·68-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.34·5-s − 0.707·8-s − 0.904·11-s + 1.66·13-s + 3/2·16-s + 2.18·17-s − 0.688·19-s + 2.01·20-s + 3·25-s − 2.22·29-s − 2.15·31-s − 1.59·32-s + 0.493·37-s − 0.948·40-s − 1.82·43-s − 1.35·44-s + 0.875·47-s + 2.49·52-s − 0.412·53-s − 1.21·55-s − 0.390·59-s − 1.92·61-s + 7/4·64-s + 2.23·65-s + 1.09·67-s + 3.27·68-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(78228.6\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.55352257\)
\(L(\frac12)\) \(\approx\) \(10.55352257\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 17 T^{3} + p^{3} T^{6} \)
13 \( ( 1 - T )^{6} \)
good2 \( 1 - 3 T^{2} + p T^{3} + 3 T^{4} - 3 T^{5} - T^{6} - 3 p T^{7} + 3 p^{2} T^{8} + p^{4} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 - 3 T - 6 T^{2} + 13 T^{3} + 63 T^{4} - 12 p T^{5} - 259 T^{6} - 12 p^{2} T^{7} + 63 p^{2} T^{8} + 13 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 24 T^{2} - 31 T^{3} + 531 T^{4} + 30 p T^{5} - 6181 T^{6} + 30 p^{2} T^{7} + 531 p^{2} T^{8} - 31 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 9 T + 6 T^{2} - 29 T^{3} + 1443 T^{4} - 228 p T^{5} - 287 p T^{6} - 228 p^{2} T^{7} + 1443 p^{2} T^{8} - 29 p^{3} T^{9} + 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 39 T^{2} - 78 T^{3} + 1059 T^{4} + 939 T^{5} - 20986 T^{6} + 939 p T^{7} + 1059 p^{2} T^{8} - 78 p^{3} T^{9} - 39 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T^{2} - 18 T^{3} - 102 T^{4} + 54 T^{5} + 23587 T^{6} + 54 p T^{7} - 102 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 6 T + 36 T^{2} + 59 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 12 T + 12 T^{2} + 58 T^{3} + 4176 T^{4} + 14040 T^{5} - 36777 T^{6} + 14040 p T^{7} + 4176 p^{2} T^{8} + 58 p^{3} T^{9} + 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 102 T^{2} + 109 T^{3} + 7551 T^{4} - 108 p T^{5} - 318051 T^{6} - 108 p^{2} T^{7} + 7551 p^{2} T^{8} + 109 p^{3} T^{9} - 102 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 96 T^{2} + 27 T^{3} + 96 p T^{4} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 6 T + 120 T^{2} + 465 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 6 T - 69 T^{2} + 450 T^{3} + 2850 T^{4} - 10734 T^{5} - 98453 T^{6} - 10734 p T^{7} + 2850 p^{2} T^{8} + 450 p^{3} T^{9} - 69 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 3 T - 150 T^{2} - 153 T^{3} + 15909 T^{4} + 150 p T^{5} - 968195 T^{6} + 150 p^{2} T^{7} + 15909 p^{2} T^{8} - 153 p^{3} T^{9} - 150 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 3 T + 48 T^{2} + 153 T^{3} + 4227 T^{4} + 29514 T^{5} + 307915 T^{6} + 29514 p T^{7} + 4227 p^{2} T^{8} + 153 p^{3} T^{9} + 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 15 T + 114 T^{2} + 19 T^{3} - 6795 T^{4} - 55962 T^{5} - 457515 T^{6} - 55962 p T^{7} - 6795 p^{2} T^{8} + 19 p^{3} T^{9} + 114 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 72 T^{2} + 889 T^{3} + 2961 T^{4} - 29232 T^{5} - 46797 T^{6} - 29232 p T^{7} + 2961 p^{2} T^{8} + 889 p^{3} T^{9} - 72 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 21 T + 267 T^{2} + 2385 T^{3} + 267 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 90 T^{2} - 142 T^{3} + 1530 T^{4} + 6390 T^{5} + 191775 T^{6} + 6390 p T^{7} + 1530 p^{2} T^{8} - 142 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 24 T + 240 T^{2} - 1522 T^{3} + 7848 T^{4} + 25272 T^{5} - 828081 T^{6} + 25272 p T^{7} + 7848 p^{2} T^{8} - 1522 p^{3} T^{9} + 240 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 12 T + 294 T^{2} - 2045 T^{3} + 294 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 15 T - 78 T^{2} + 617 T^{3} + 26127 T^{4} - 1032 p T^{5} - 21143 p T^{6} - 1032 p^{2} T^{7} + 26127 p^{2} T^{8} + 617 p^{3} T^{9} - 78 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 21 T + 375 T^{2} - 3805 T^{3} + 375 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.53512841598368733770321757681, −5.32001252359010376400679248928, −5.28198317622227758081758285631, −5.14362230866252066859423335310, −4.88267001575096858109643153482, −4.52100962622302400690558776906, −4.45699090859527925917567508907, −4.44256126107347988722594541220, −3.96167383140057445458880379469, −3.79348466323644940347311047065, −3.53361197551769825157918973719, −3.30152091762462332257994884821, −3.24055375960637549859700285835, −3.20787331899918717446617096992, −3.15471256839176610585317496622, −2.84645808086428443145064051981, −2.48886052745411542826786170203, −2.12010148606258489383568197219, −2.00964114742948067106068979970, −1.91179211623239989375340910036, −1.81181729600691819438022969705, −1.42478043098986362299068235094, −1.23433213916247244576929954729, −0.67070803828216033536479638577, −0.56269624482262792310325019266, 0.56269624482262792310325019266, 0.67070803828216033536479638577, 1.23433213916247244576929954729, 1.42478043098986362299068235094, 1.81181729600691819438022969705, 1.91179211623239989375340910036, 2.00964114742948067106068979970, 2.12010148606258489383568197219, 2.48886052745411542826786170203, 2.84645808086428443145064051981, 3.15471256839176610585317496622, 3.20787331899918717446617096992, 3.24055375960637549859700285835, 3.30152091762462332257994884821, 3.53361197551769825157918973719, 3.79348466323644940347311047065, 3.96167383140057445458880379469, 4.44256126107347988722594541220, 4.45699090859527925917567508907, 4.52100962622302400690558776906, 4.88267001575096858109643153482, 5.14362230866252066859423335310, 5.28198317622227758081758285631, 5.32001252359010376400679248928, 5.53512841598368733770321757681

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

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