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Properties

Label 12-3e30-1.1-c1e6-0-1
Degree $12$
Conductor $2.059\times 10^{14}$
Sign $1$
Analytic cond. $53.3703$
Root an. cond. $1.39296$
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 + 6·4-s + 6·5-s + 3·7-s + 9·8-s + 18·10-s + 3·11-s + 3·13-s + 9·14-s + 12·16-s − 18·17-s − 6·19-s + 36·20-s + 9·22-s + 6·23-s + 24·25-s + 9·26-s + 18·28-s + 12·29-s + 12·31-s + 12·32-s − 54·34-s + 18·35-s − 6·37-s − 18·38-s + 54·40-s − 3·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 2.68·5-s + 1.13·7-s + 3.18·8-s + 5.69·10-s + 0.904·11-s + 0.832·13-s + 2.40·14-s + 3·16-s − 4.36·17-s − 1.37·19-s + 8.04·20-s + 1.91·22-s + 1.25·23-s + 24/5·25-s + 1.76·26-s + 3.40·28-s + 2.22·29-s + 2.15·31-s + 2.12·32-s − 9.26·34-s + 3.04·35-s − 0.986·37-s − 2.91·38-s + 8.53·40-s − 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{30}\)
Sign: $1$
Analytic conductor: \(53.3703\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{30} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.58576624\)
\(L(\frac12)\) \(\approx\) \(17.58576624\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - 11 T^{6} + 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 6 T + 12 T^{2} - 18 T^{3} + 78 T^{4} - 222 T^{5} + 439 T^{6} - 222 p T^{7} + 78 p^{2} T^{8} - 18 p^{3} T^{9} + 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T - 6 T^{2} + 5 T^{3} + 45 T^{4} + 108 T^{5} - 705 T^{6} + 108 p T^{7} + 45 p^{2} T^{8} + 5 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T - 6 T^{2} + 81 T^{3} - 129 T^{4} - 318 T^{5} + 3067 T^{6} - 318 p T^{7} - 129 p^{2} T^{8} + 81 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T - 24 T^{2} + 23 T^{3} + 477 T^{4} + 54 T^{5} - 7563 T^{6} + 54 p T^{7} + 477 p^{2} T^{8} + 23 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 3 T + p T^{2} )^{6} \)
19 \( ( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 6 T - 24 T^{2} + 90 T^{3} + 870 T^{4} - 114 T^{5} - 26885 T^{6} - 114 p T^{7} + 870 p^{2} T^{8} + 90 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 12 T + 30 T^{2} - 90 T^{3} + 2022 T^{4} - 3594 T^{5} - 40889 T^{6} - 3594 p T^{7} + 2022 p^{2} T^{8} - 90 p^{3} T^{9} + 30 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 3 T - 60 T^{2} + 153 T^{3} + 1851 T^{4} - 10068 T^{5} - 71759 T^{6} - 10068 p T^{7} + 1851 p^{2} T^{8} + 153 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 12 T - 24 T^{2} + 2 p T^{3} + 8388 T^{4} - 25380 T^{5} - 216273 T^{6} - 25380 p T^{7} + 8388 p^{2} T^{8} + 2 p^{4} T^{9} - 24 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T - 42 T^{2} - 126 T^{3} + 636 T^{4} - 7572 T^{5} - 69617 T^{6} - 7572 p T^{7} + 636 p^{2} T^{8} - 126 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 21 T + 120 T^{2} + 1143 T^{3} + 25323 T^{4} + 176142 T^{5} + 630619 T^{6} + 176142 p T^{7} + 25323 p^{2} T^{8} + 1143 p^{3} T^{9} + 120 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 96 T^{2} - 778 T^{3} + 4662 T^{4} + 29322 T^{5} - 126633 T^{6} + 29322 p T^{7} + 4662 p^{2} T^{8} - 778 p^{3} T^{9} - 96 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 114 T^{2} - 490 T^{3} + 8280 T^{4} + 10584 T^{5} - 584553 T^{6} + 10584 p T^{7} + 8280 p^{2} T^{8} - 490 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 51 T^{2} - 279 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 6 T - 150 T^{2} - 886 T^{3} + 13896 T^{4} + 47304 T^{5} - 969681 T^{6} + 47304 p T^{7} + 13896 p^{2} T^{8} - 886 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 30012 p T^{7} + 25188 p^{2} T^{8} - 558 p^{3} T^{9} - 186 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 78 T^{2} - 999 T^{3} + 78 p T^{4} + p^{3} T^{6} )^{2} \)
97 \( 1 + 15 T + 3 T^{2} - 2452 T^{3} - 15615 T^{4} + 136917 T^{5} + 3285654 T^{6} + 136917 p T^{7} - 15615 p^{2} T^{8} - 2452 p^{3} T^{9} + 3 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.59099089089800989513481856479, −6.43126229866415703057915634800, −6.22629403480118035471323786931, −6.18278929059991075745281497434, −6.11211145345343714273002250432, −5.49869552792536264479224957298, −5.49214105282758940719463297645, −5.43983327985074683071807670461, −4.88939337116370981639806746915, −4.81529407614241932614476437020, −4.74564252061874689332803349839, −4.56314641516219728333431657789, −4.30572697148041977933841722946, −4.28919918269374771642342202537, −4.06257227749252560016882852512, −3.55766295003154750395776900164, −3.21330485047285602568872488232, −2.94557159852271556906797993727, −2.83621914765407178998860006995, −2.43165677394746859561279716584, −2.42459857772813078991917157182, −2.09821743474737960423295169557, −1.66853619473493056193113358103, −1.43242677796416715617369024414, −1.30247530293544449143022274277, 1.30247530293544449143022274277, 1.43242677796416715617369024414, 1.66853619473493056193113358103, 2.09821743474737960423295169557, 2.42459857772813078991917157182, 2.43165677394746859561279716584, 2.83621914765407178998860006995, 2.94557159852271556906797993727, 3.21330485047285602568872488232, 3.55766295003154750395776900164, 4.06257227749252560016882852512, 4.28919918269374771642342202537, 4.30572697148041977933841722946, 4.56314641516219728333431657789, 4.74564252061874689332803349839, 4.81529407614241932614476437020, 4.88939337116370981639806746915, 5.43983327985074683071807670461, 5.49214105282758940719463297645, 5.49869552792536264479224957298, 6.11211145345343714273002250432, 6.18278929059991075745281497434, 6.22629403480118035471323786931, 6.43126229866415703057915634800, 6.59099089089800989513481856479

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

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