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Properties

Label 12-19e12-1.1-c1e6-0-9
Degree $12$
Conductor $2.213\times 10^{15}$
Sign $1$
Analytic cond. $573.727$
Root an. cond. $1.69782$
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  + 3·2-s + 6·3-s + 9·4-s + 3·5-s + 18·6-s + 18·8-s + 24·9-s + 9·10-s + 54·12-s − 12·13-s + 18·15-s + 36·16-s − 6·17-s + 72·18-s + 27·20-s − 12·23-s + 108·24-s + 9·25-s − 36·26-s + 71·27-s − 3·29-s + 54·30-s + 9·31-s + 66·32-s − 18·34-s + 216·36-s − 72·39-s + ⋯
L(s)  = 1  + 2.12·2-s + 3.46·3-s + 9/2·4-s + 1.34·5-s + 7.34·6-s + 6.36·8-s + 8·9-s + 2.84·10-s + 15.5·12-s − 3.32·13-s + 4.64·15-s + 9·16-s − 1.45·17-s + 16.9·18-s + 6.03·20-s − 2.50·23-s + 22.0·24-s + 9/5·25-s − 7.06·26-s + 13.6·27-s − 0.557·29-s + 9.85·30-s + 1.61·31-s + 11.6·32-s − 3.08·34-s + 36·36-s − 11.5·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(19^{12}\)
Sign: $1$
Analytic conductor: \(573.727\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(83.15061945\)
\(L(\frac12)\) \(\approx\) \(83.15061945\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 - 3 T + 9 T^{3} - 9 T^{4} - 3 p^{2} T^{5} + 37 T^{6} - 3 p^{3} T^{7} - 9 p^{2} T^{8} + 9 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 - 2 p T + 4 p T^{2} + T^{3} - 13 p T^{4} + 17 p T^{5} - 35 T^{6} + 17 p^{2} T^{7} - 13 p^{3} T^{8} + p^{3} T^{9} + 4 p^{5} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 18 T^{3} - 36 T^{4} - 3 p^{2} T^{5} + 379 T^{6} - 3 p^{3} T^{7} - 36 p^{2} T^{8} + 18 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 18 T^{2} + 2 T^{3} + 198 T^{4} - 18 T^{5} - 1581 T^{6} - 18 p T^{7} + 198 p^{2} T^{8} + 2 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 216 p T^{7} + 312 p^{2} T^{8} - 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 12 T + 69 T^{2} + 215 T^{3} + 45 T^{4} - 3303 T^{5} - 17910 T^{6} - 3303 p T^{7} + 45 p^{2} T^{8} + 215 p^{3} T^{9} + 69 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T + 9 T^{2} - 99 T^{3} - 423 T^{4} - 435 T^{5} + 2746 T^{6} - 435 p T^{7} - 423 p^{2} T^{8} - 99 p^{3} T^{9} + 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 72 T^{2} + 378 T^{3} + 1404 T^{4} + 2658 T^{5} + 4969 T^{6} + 2658 p T^{7} + 1404 p^{2} T^{8} + 378 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T - 9 T^{2} + 243 T^{3} - 564 T^{5} + 64405 T^{6} - 564 p T^{7} + 243 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 3402 p T^{7} + 2187 p^{2} T^{8} + 119 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 90 T^{2} - 17 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 198 T^{2} + 1980 T^{3} + 18585 T^{4} + 138093 T^{5} + 959257 T^{6} + 138093 p T^{7} + 18585 p^{2} T^{8} + 1980 p^{3} T^{9} + 198 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 21 T + 204 T^{2} + 1034 T^{3} - 1035 T^{4} - 73071 T^{5} - 686457 T^{6} - 73071 p T^{7} - 1035 p^{2} T^{8} + 1034 p^{3} T^{9} + 204 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 12 T + 63 T^{2} + 333 T^{3} - 1233 T^{4} - 36915 T^{5} - 283166 T^{6} - 36915 p T^{7} - 1233 p^{2} T^{8} + 333 p^{3} T^{9} + 63 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 15 T + 90 T^{2} + 18 T^{3} - 3447 T^{4} + 44751 T^{5} - 378575 T^{6} + 44751 p T^{7} - 3447 p^{2} T^{8} + 18 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 108 T^{2} + 1296 T^{3} + 12006 T^{4} + 92490 T^{5} + 996409 T^{6} + 92490 p T^{7} + 12006 p^{2} T^{8} + 1296 p^{3} T^{9} + 108 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 132 T^{2} + 710 T^{3} + 6768 T^{4} + 15840 T^{5} + 182943 T^{6} + 15840 p T^{7} + 6768 p^{2} T^{8} + 710 p^{3} T^{9} + 132 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 24 T + 456 T^{2} - 6454 T^{3} + 78804 T^{4} - 795726 T^{5} + 7018533 T^{6} - 795726 p T^{7} + 78804 p^{2} T^{8} - 6454 p^{3} T^{9} + 456 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T + 1458 T^{3} - 9540 T^{4} - 73218 T^{5} + 1401913 T^{6} - 73218 p T^{7} - 9540 p^{2} T^{8} + 1458 p^{3} T^{9} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T + 96 T^{2} + 512 T^{3} - 432 T^{4} - 79704 T^{5} - 815913 T^{6} - 79704 p T^{7} - 432 p^{2} T^{8} + 512 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 15 T + 87 T^{2} + 179 T^{3} - 1674 T^{4} - 76626 T^{5} + 1419201 T^{6} - 76626 p T^{7} - 1674 p^{2} T^{8} + 179 p^{3} T^{9} + 87 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 60 T^{2} + 918 T^{3} - 1380 T^{4} - 27540 T^{5} + 1055455 T^{6} - 27540 p T^{7} - 1380 p^{2} T^{8} + 918 p^{3} T^{9} - 60 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 15 T + 36 T^{2} + 1548 T^{3} - 15786 T^{4} - 7827 T^{5} + 1123759 T^{6} - 7827 p T^{7} - 15786 p^{2} T^{8} + 1548 p^{3} T^{9} + 36 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 54 T^{2} - 1042 T^{3} - 9018 T^{4} + 111078 T^{5} + 1328151 T^{6} + 111078 p T^{7} - 9018 p^{2} T^{8} - 1042 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} \)
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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

−6.37847706781013763753818904233, −6.25201268874183161977803209055, −5.78956151395153410023383275614, −5.66456303312131575125555969198, −5.39103328404125706719420891092, −5.11049851311927378754126506968, −5.07409692822384995791848467623, −4.88425513517925634903157993464, −4.58595991111283794719373169236, −4.50684818966749780723700394509, −4.26579434736974276958868896080, −4.22387579270721921685126152416, −3.67035848280779182973914119948, −3.47889058210339725863452411004, −3.44667010849219289029932484472, −3.43577086366452712680440356581, −2.93852847792827753103953284894, −2.63275422941151613120831954990, −2.44566335777261091915037054167, −2.40387679045036001105541194947, −2.35973267346826351318301140569, −1.88907822640327879056418682503, −1.87420826775598181020103206031, −1.76121628343968130038496707340, −1.17431357306883297853017935726, 1.17431357306883297853017935726, 1.76121628343968130038496707340, 1.87420826775598181020103206031, 1.88907822640327879056418682503, 2.35973267346826351318301140569, 2.40387679045036001105541194947, 2.44566335777261091915037054167, 2.63275422941151613120831954990, 2.93852847792827753103953284894, 3.43577086366452712680440356581, 3.44667010849219289029932484472, 3.47889058210339725863452411004, 3.67035848280779182973914119948, 4.22387579270721921685126152416, 4.26579434736974276958868896080, 4.50684818966749780723700394509, 4.58595991111283794719373169236, 4.88425513517925634903157993464, 5.07409692822384995791848467623, 5.11049851311927378754126506968, 5.39103328404125706719420891092, 5.66456303312131575125555969198, 5.78956151395153410023383275614, 6.25201268874183161977803209055, 6.37847706781013763753818904233

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

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