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Properties

Label 12-1710e6-1.1-c1e6-0-2
Degree $12$
Conductor $2.500\times 10^{19}$
Sign $1$
Analytic cond. $6.48095\times 10^{6}$
Root an. cond. $3.69518$
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 + 3·4-s + 3·5-s + 2·7-s − 2·8-s + 9·10-s − 8·11-s − 13-s + 6·14-s − 9·16-s − 4·17-s − 2·19-s + 9·20-s − 24·22-s − 4·23-s + 3·25-s − 3·26-s + 6·28-s + 6·29-s + 2·31-s − 9·32-s − 12·34-s + 6·35-s − 6·37-s − 6·38-s − 6·40-s + 8·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s + 1.34·5-s + 0.755·7-s − 0.707·8-s + 2.84·10-s − 2.41·11-s − 0.277·13-s + 1.60·14-s − 9/4·16-s − 0.970·17-s − 0.458·19-s + 2.01·20-s − 5.11·22-s − 0.834·23-s + 3/5·25-s − 0.588·26-s + 1.13·28-s + 1.11·29-s + 0.359·31-s − 1.59·32-s − 2.05·34-s + 1.01·35-s − 0.986·37-s − 0.973·38-s − 0.948·40-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(6.48095\times 10^{6}\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.114937781\)
\(L(\frac12)\) \(\approx\) \(4.114937781\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
19 \( 1 + 2 T + 8 T^{2} + 140 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( ( 1 - T + 2 T^{2} + 17 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 4 T + 2 p T^{2} + 82 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + T - 22 T^{2} - 45 T^{3} + 196 T^{4} + 353 T^{5} - 1496 T^{6} + 353 p T^{7} + 196 p^{2} T^{8} - 45 p^{3} T^{9} - 22 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 15 T^{2} + 164 T^{3} + 304 T^{4} + 940 T^{5} + 11585 T^{6} + 940 p T^{7} + 304 p^{2} T^{8} + 164 p^{3} T^{9} + 15 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T - 42 T^{2} - 124 T^{3} + 1318 T^{4} + 1768 T^{5} - 29818 T^{6} + 1768 p T^{7} + 1318 p^{2} T^{8} - 124 p^{3} T^{9} - 42 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 51 T^{2} - 582 T^{3} + 2676 T^{4} - 15306 T^{5} + 124873 T^{6} - 15306 p T^{7} + 2676 p^{2} T^{8} - 582 p^{3} T^{9} + 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - T + 77 T^{2} - 70 T^{3} + 77 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + T + p T^{2} )^{6} \)
41 \( 1 - 8 T + 66 T^{2} - 748 T^{3} + 3142 T^{4} - 15524 T^{5} + 176018 T^{6} - 15524 p T^{7} + 3142 p^{2} T^{8} - 748 p^{3} T^{9} + 66 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - 14 T + 48 T^{2} + 44 T^{3} + 48 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )( 1 + 25 T + 318 T^{2} + 2543 T^{3} + 318 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( 1 - 57 T^{2} - 288 T^{3} + 570 T^{4} + 8208 T^{5} + 75679 T^{6} + 8208 p T^{7} + 570 p^{2} T^{8} - 288 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 18 T + 78 T^{2} - 396 T^{3} + 11190 T^{4} - 77130 T^{5} + 244546 T^{6} - 77130 p T^{7} + 11190 p^{2} T^{8} - 396 p^{3} T^{9} + 78 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \)
61 \( 1 - 3 T - 156 T^{2} + 233 T^{3} + 16068 T^{4} - 10833 T^{5} - 1107756 T^{6} - 10833 p T^{7} + 16068 p^{2} T^{8} + 233 p^{3} T^{9} - 156 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 15 T - 30 T^{2} - 199 T^{3} + 17760 T^{4} + 67845 T^{5} - 717030 T^{6} + 67845 p T^{7} + 17760 p^{2} T^{8} - 199 p^{3} T^{9} - 30 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 4 T - 147 T^{2} - 52 T^{3} + 13588 T^{4} - 15044 T^{5} - 1127329 T^{6} - 15044 p T^{7} + 13588 p^{2} T^{8} - 52 p^{3} T^{9} - 147 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 21 T + 96 T^{2} - 685 T^{3} + 24960 T^{4} - 185739 T^{5} + 497520 T^{6} - 185739 p T^{7} + 24960 p^{2} T^{8} - 685 p^{3} T^{9} + 96 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T - 188 T^{2} + 529 T^{3} + 29788 T^{4} - 517 p T^{5} - 2560246 T^{6} - 517 p^{2} T^{7} + 29788 p^{2} T^{8} + 529 p^{3} T^{9} - 188 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 2 T - 11 T^{2} - 700 T^{3} - 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 18 T - 30 T^{2} - 252 T^{3} + 34194 T^{4} + 180810 T^{5} - 1340246 T^{6} + 180810 p T^{7} + 34194 p^{2} T^{8} - 252 p^{3} T^{9} - 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 38 T + 691 T^{2} + 10246 T^{3} + 144976 T^{4} + 1721978 T^{5} + 17584733 T^{6} + 1721978 p T^{7} + 144976 p^{2} T^{8} + 10246 p^{3} T^{9} + 691 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.98459104588082775690697021182, −4.74157421075679348500260063276, −4.56836485964691076006774213762, −4.43050879506436436806433867890, −4.20233924723427914134508508438, −4.10466216660296874382364951943, −4.09195758075160160462586400485, −4.05629774560629155614909858357, −3.71561918262381216667853377636, −3.36303570407595423943006407148, −3.30761891089830727333474717912, −3.26256506819235183602423474582, −2.75022171422650488970917553252, −2.73404266841007367065820807432, −2.67955361423809943975507805962, −2.46749543500379366923860235979, −2.42193136415198379481636951675, −2.06007310842979834942074508299, −1.93273599126173492818687896023, −1.93163863113507336132041651264, −1.31216249679929664331950819900, −1.22506868499457810535065705296, −1.07441655089370514084270165518, −0.36321989711770345202330099405, −0.23015875231207545739589979128, 0.23015875231207545739589979128, 0.36321989711770345202330099405, 1.07441655089370514084270165518, 1.22506868499457810535065705296, 1.31216249679929664331950819900, 1.93163863113507336132041651264, 1.93273599126173492818687896023, 2.06007310842979834942074508299, 2.42193136415198379481636951675, 2.46749543500379366923860235979, 2.67955361423809943975507805962, 2.73404266841007367065820807432, 2.75022171422650488970917553252, 3.26256506819235183602423474582, 3.30761891089830727333474717912, 3.36303570407595423943006407148, 3.71561918262381216667853377636, 4.05629774560629155614909858357, 4.09195758075160160462586400485, 4.10466216660296874382364951943, 4.20233924723427914134508508438, 4.43050879506436436806433867890, 4.56836485964691076006774213762, 4.74157421075679348500260063276, 4.98459104588082775690697021182

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

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