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Properties

Label 12-1008e6-1.1-c1e6-0-8
Degree $12$
Conductor $1.049\times 10^{18}$
Sign $1$
Analytic cond. $271910.$
Root an. cond. $2.83706$
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  + 3·5-s + 3·7-s + 3·13-s + 12·17-s + 18·19-s + 6·23-s + 15·25-s + 9·27-s + 3·29-s − 9·31-s + 9·35-s − 30·37-s − 6·41-s − 15·43-s + 9·47-s + 3·49-s − 12·53-s + 3·59-s + 12·61-s + 9·65-s − 12·67-s − 42·71-s + 6·73-s − 15·79-s − 6·83-s + 36·85-s − 36·89-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s + 0.832·13-s + 2.91·17-s + 4.12·19-s + 1.25·23-s + 3·25-s + 1.73·27-s + 0.557·29-s − 1.61·31-s + 1.52·35-s − 4.93·37-s − 0.937·41-s − 2.28·43-s + 1.31·47-s + 3/7·49-s − 1.64·53-s + 0.390·59-s + 1.53·61-s + 1.11·65-s − 1.46·67-s − 4.98·71-s + 0.702·73-s − 1.68·79-s − 0.658·83-s + 3.90·85-s − 3.81·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(271910.\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.39988865\)
\(L(\frac12)\) \(\approx\) \(16.39988865\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
7 \( ( 1 - T + T^{2} )^{3} \)
good5 \( 1 - 3 T - 6 T^{2} + 13 T^{3} + 63 T^{4} - 12 p T^{5} - 259 T^{6} - 12 p^{2} T^{7} + 63 p^{2} T^{8} + 13 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} + 18 T^{3} + 312 T^{4} - 216 T^{5} - 3593 T^{6} - 216 p T^{7} + 312 p^{2} T^{8} + 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 3 T - 21 T^{2} + 60 T^{3} + 285 T^{4} - 417 T^{5} - 3202 T^{6} - 417 p T^{7} + 285 p^{2} T^{8} + 60 p^{3} T^{9} - 21 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 6 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 9 T + 63 T^{2} - 17 p T^{3} + 63 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 6 T + 12 T^{2} - 130 T^{3} - 18 T^{4} + 3282 T^{5} - 7909 T^{6} + 3282 p T^{7} - 18 p^{2} T^{8} - 130 p^{3} T^{9} + 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 3 T - 42 T^{2} + 157 T^{3} + 657 T^{4} - 1554 T^{5} - 10195 T^{6} - 1554 p T^{7} + 657 p^{2} T^{8} + 157 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T - 18 T^{2} - 79 T^{3} + 2367 T^{4} + 1422 T^{5} - 86865 T^{6} + 1422 p T^{7} + 2367 p^{2} T^{8} - 79 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 165 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 6 T - 72 T^{2} - 298 T^{3} + 4398 T^{4} + 8034 T^{5} - 177169 T^{6} + 8034 p T^{7} + 4398 p^{2} T^{8} - 298 p^{3} T^{9} - 72 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 42 T^{2} + 199 T^{3} + 6525 T^{4} + 24624 T^{5} - 76509 T^{6} + 24624 p T^{7} + 6525 p^{2} T^{8} + 199 p^{3} T^{9} + 42 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T - 66 T^{2} + 371 T^{3} + 7077 T^{4} - 20028 T^{5} - 269497 T^{6} - 20028 p T^{7} + 7077 p^{2} T^{8} + 371 p^{3} T^{9} - 66 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 162 T^{2} + 617 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 24 T^{2} - 369 T^{3} - 453 T^{4} + 5694 T^{5} + 333403 T^{6} + 5694 p T^{7} - 453 p^{2} T^{8} - 369 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T - 48 T^{2} + 478 T^{3} + 8460 T^{4} - 30636 T^{5} - 395685 T^{6} - 30636 p T^{7} + 8460 p^{2} T^{8} + 478 p^{3} T^{9} - 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 12 T - 12 T^{2} - 690 T^{3} - 1488 T^{4} + 2856 T^{5} - 34441 T^{6} + 2856 p T^{7} - 1488 p^{2} T^{8} - 690 p^{3} T^{9} - 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 21 T + 249 T^{2} + 2115 T^{3} + 249 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 3 T + 105 T^{2} - 475 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 15 T + 30 T^{2} + 59 T^{3} + 1125 T^{4} - 89100 T^{5} - 1368633 T^{6} - 89100 p T^{7} + 1125 p^{2} T^{8} + 59 p^{3} T^{9} + 30 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 42 T^{2} - 2610 T^{3} - 9624 T^{4} + 83688 T^{5} + 2759119 T^{6} + 83688 p T^{7} - 9624 p^{2} T^{8} - 2610 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 18 T + 318 T^{2} + 3241 T^{3} + 318 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 15 T + 168 T^{2} + 1477 T^{3} + 2385 T^{4} - 42462 T^{5} - 510111 T^{6} - 42462 p T^{7} + 2385 p^{2} T^{8} + 1477 p^{3} T^{9} + 168 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.39030695747320763867833658874, −5.15322549965987162379983877547, −5.07941971579045340154280058969, −4.81013358089598661199902571428, −4.67671922338897565095833053415, −4.66042793191578956307680875446, −4.42781065817627612562104314439, −4.12906842990916666104692781190, −3.96361638528571575286311004204, −3.54631601756940740412196301458, −3.35651725220645905368586818790, −3.16158505888743613240350162483, −3.15562331937802466851616446493, −3.15342914830882326567102860683, −3.14979542044860197582821434827, −2.74343445856283867004948612253, −2.68634789574734476642563894071, −1.96972346290567967426174277039, −1.74668054940701417677491024070, −1.61372240167283562437948452942, −1.58106444922719326418565708063, −1.31121132732396037901066303278, −1.27412356874032592450559692955, −0.73731417234963549790849247879, −0.60438639838652310942737813186, 0.60438639838652310942737813186, 0.73731417234963549790849247879, 1.27412356874032592450559692955, 1.31121132732396037901066303278, 1.58106444922719326418565708063, 1.61372240167283562437948452942, 1.74668054940701417677491024070, 1.96972346290567967426174277039, 2.68634789574734476642563894071, 2.74343445856283867004948612253, 3.14979542044860197582821434827, 3.15342914830882326567102860683, 3.15562331937802466851616446493, 3.16158505888743613240350162483, 3.35651725220645905368586818790, 3.54631601756940740412196301458, 3.96361638528571575286311004204, 4.12906842990916666104692781190, 4.42781065817627612562104314439, 4.66042793191578956307680875446, 4.67671922338897565095833053415, 4.81013358089598661199902571428, 5.07941971579045340154280058969, 5.15322549965987162379983877547, 5.39030695747320763867833658874

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

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