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Properties

Label 12-338e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.491\times 10^{15}$
Sign $1$
Analytic cond. $386.511$
Root an. cond. $1.64284$
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 + 3·4-s + 4·5-s + 9·6-s + 4·7-s + 2·8-s + 5·9-s − 12·10-s − 3·11-s − 9·12-s − 12·14-s − 12·15-s − 9·16-s − 5·17-s − 15·18-s − 19-s + 12·20-s − 12·21-s + 9·22-s − 6·24-s − 2·25-s − 8·27-s + 12·28-s + 10·29-s + 36·30-s − 32·31-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3/2·4-s + 1.78·5-s + 3.67·6-s + 1.51·7-s + 0.707·8-s + 5/3·9-s − 3.79·10-s − 0.904·11-s − 2.59·12-s − 3.20·14-s − 3.09·15-s − 9/4·16-s − 1.21·17-s − 3.53·18-s − 0.229·19-s + 2.68·20-s − 2.61·21-s + 1.91·22-s − 1.22·24-s − 2/5·25-s − 1.53·27-s + 2.26·28-s + 1.85·29-s + 6.57·30-s − 5.74·31-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(386.511\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1010647445\)
\(L(\frac12)\) \(\approx\) \(0.1010647445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
13 \( 1 \)
good3 \( 1 + p T + 4 T^{2} + 5 T^{3} - 5 T^{4} - 38 T^{5} - 77 T^{6} - 38 p T^{7} - 5 p^{2} T^{8} + 5 p^{3} T^{9} + 4 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 4 T - T^{2} + 4 p T^{3} - 22 T^{4} - 60 T^{5} + 183 T^{6} - 60 p T^{7} - 22 p^{2} T^{8} + 4 p^{4} T^{9} - p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 20 T^{2} - 19 T^{3} + 3 p^{2} T^{4} - 14 T^{5} - 4869 T^{6} - 14 p T^{7} + 3 p^{4} T^{8} - 19 p^{3} T^{9} - 20 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 5 T - 4 T^{2} - T^{3} - p T^{4} - 1368 T^{5} - 6367 T^{6} - 1368 p T^{7} - p^{3} T^{8} - p^{3} T^{9} - 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + T - 40 T^{2} - 61 T^{3} + 851 T^{4} + 894 T^{5} - 15981 T^{6} + 894 p T^{7} + 851 p^{2} T^{8} - 61 p^{3} T^{9} - 40 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 41 T^{2} + 112 T^{3} + 738 T^{4} - 2296 T^{5} - 11193 T^{6} - 2296 p T^{7} + 738 p^{2} T^{8} + 112 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 10 T + 17 T^{2} + 122 T^{3} - 278 T^{4} + 222 T^{5} - 8527 T^{6} + 222 p T^{7} - 278 p^{2} T^{8} + 122 p^{3} T^{9} + 17 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 16 T + 169 T^{2} + 1096 T^{3} + 169 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 14 T + 29 T^{2} - 154 T^{3} + 6422 T^{4} - 28518 T^{5} - 6447 T^{6} - 28518 p T^{7} + 6422 p^{2} T^{8} - 154 p^{3} T^{9} + 29 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 7 T - 60 T^{2} - 203 T^{3} + 4031 T^{4} + 1848 T^{5} - 198359 T^{6} + 1848 p T^{7} + 4031 p^{2} T^{8} - 203 p^{3} T^{9} - 60 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 11 T - 4 T^{2} - 515 T^{3} - 145 T^{4} + 20214 T^{5} + 156435 T^{6} + 20214 p T^{7} - 145 p^{2} T^{8} - 515 p^{3} T^{9} - 4 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 2 T + 105 T^{2} + 180 T^{3} + 105 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 131 T^{2} - 56 T^{3} + 131 p T^{4} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 108 T^{2} + 611 T^{3} + 5343 T^{4} - 25302 T^{5} - 196861 T^{6} - 25302 p T^{7} + 5343 p^{2} T^{8} + 611 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T - 23 T^{2} + 692 T^{3} - 2554 T^{4} - 12388 T^{5} + 481421 T^{6} - 12388 p T^{7} - 2554 p^{2} T^{8} + 692 p^{3} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 21 T + 156 T^{2} + 931 T^{3} + 9063 T^{4} + 12474 T^{5} - 434469 T^{6} + 12474 p T^{7} + 9063 p^{2} T^{8} + 931 p^{3} T^{9} + 156 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T - 89 T^{2} + 628 T^{3} + 17202 T^{4} - 62356 T^{5} - 933657 T^{6} - 62356 p T^{7} + 17202 p^{2} T^{8} + 628 p^{3} T^{9} - 89 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - T + 133 T^{2} - 397 T^{3} + 133 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + T + 233 T^{2} + 179 T^{3} + 233 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 25 T + 264 T^{2} + 1639 T^{3} + 5867 T^{4} - 75252 T^{5} - 1501007 T^{6} - 75252 p T^{7} + 5867 p^{2} T^{8} + 1639 p^{3} T^{9} + 264 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 23 T + 176 T^{2} + 961 T^{3} + 83 p T^{4} - 95292 T^{5} - 2429847 T^{6} - 95292 p T^{7} + 83 p^{3} T^{8} + 961 p^{3} T^{9} + 176 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.38670363995462338998216423175, −6.01600840931380792474115826700, −5.86799428342797826067034954372, −5.73686732832643298276357952780, −5.49884868430415157924867057342, −5.47480828643883784025301631126, −5.31275359542175198000225715849, −5.17783492135214138652498778721, −4.93363652332603313301164947944, −4.60142643629025004703552239143, −4.37976511071894739692784579219, −4.27830883471772750178638811365, −4.27761746080295665874480269363, −3.75978585929932131137859581796, −3.70003615021740839695636772543, −3.07889329322552098974641352159, −3.07159819627641704994106826823, −2.52376663183322469146357626148, −2.27473069989448870757423166664, −1.89489794545941285226224055348, −1.80460199138052354226772363009, −1.68535798131608311633509970104, −1.47598106817497577183703762647, −0.78219426093551626854111152313, −0.19928479515654263867817178747, 0.19928479515654263867817178747, 0.78219426093551626854111152313, 1.47598106817497577183703762647, 1.68535798131608311633509970104, 1.80460199138052354226772363009, 1.89489794545941285226224055348, 2.27473069989448870757423166664, 2.52376663183322469146357626148, 3.07159819627641704994106826823, 3.07889329322552098974641352159, 3.70003615021740839695636772543, 3.75978585929932131137859581796, 4.27761746080295665874480269363, 4.27830883471772750178638811365, 4.37976511071894739692784579219, 4.60142643629025004703552239143, 4.93363652332603313301164947944, 5.17783492135214138652498778721, 5.31275359542175198000225715849, 5.47480828643883784025301631126, 5.49884868430415157924867057342, 5.73686732832643298276357952780, 5.86799428342797826067034954372, 6.01600840931380792474115826700, 6.38670363995462338998216423175

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

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