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Properties

Label 12-126e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.002\times 10^{12}$
Sign $1$
Analytic cond. $1.03725$
Root an. cond. $1.00305$
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 + 4·3-s + 3·4-s + 10·5-s − 12·6-s − 2·7-s + 2·8-s + 6·9-s − 30·10-s + 2·11-s + 12·12-s − 2·13-s + 6·14-s + 40·15-s − 9·16-s − 4·17-s − 18·18-s − 3·19-s + 30·20-s − 8·21-s − 6·22-s + 14·23-s + 8·24-s + 37·25-s + 6·26-s + 5·27-s − 6·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 2.30·3-s + 3/2·4-s + 4.47·5-s − 4.89·6-s − 0.755·7-s + 0.707·8-s + 2·9-s − 9.48·10-s + 0.603·11-s + 3.46·12-s − 0.554·13-s + 1.60·14-s + 10.3·15-s − 9/4·16-s − 0.970·17-s − 4.24·18-s − 0.688·19-s + 6.70·20-s − 1.74·21-s − 1.27·22-s + 2.91·23-s + 1.63·24-s + 37/5·25-s + 1.17·26-s + 0.962·27-s − 1.13·28-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.03725\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.434078450\)
\(L(\frac12)\) \(\approx\) \(1.434078450\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + 10 T^{2} - 7 p T^{3} + 10 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 + 2 T + 2 T^{2} - 19 T^{3} + 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( ( 1 - p T + 19 T^{2} - 47 T^{3} + 19 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - T + 7 T^{2} - 5 p T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 2 T - 32 T^{2} - 2 p T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 230 p T^{7} + 730 p^{2} T^{8} - 2 p^{4} T^{9} - 32 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 7 T + 73 T^{2} - 319 T^{3} + 73 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 6020 p T^{7} - 185 p^{2} T^{8} - 371 p^{3} T^{9} - 30 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 14 T + 58 T^{2} + 250 T^{3} + 2992 T^{4} + 9728 T^{5} - 11857 T^{6} + 9728 p T^{7} + 2992 p^{2} T^{8} + 250 p^{3} T^{9} + 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 24546 p T^{7} + 7470 p^{2} T^{8} - 78 p^{3} T^{9} - 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 9786 p T^{7} + 7263 p^{2} T^{8} + 267 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 26334 p T^{7} - 1179 p^{2} T^{8} + 873 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 464 p T^{5} + 7381 p T^{6} - 464 p^{2} T^{7} - 1304 p^{2} T^{8} + 994 p^{3} T^{9} - 60 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 19532 p T^{7} + 292 p^{2} T^{8} - 650 p^{3} T^{9} - 32 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 32420 p T^{7} + 29404 p^{2} T^{8} + 518 p^{3} T^{9} - 180 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 545924 p T^{7} + 59506 p^{2} T^{8} + 2820 p^{3} T^{9} + 257 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.67823653668736374535779847882, −7.37866297649901416844393703707, −7.23632067042690379541667706123, −6.87986371331111101315767090232, −6.84586009112041544798600984082, −6.49010363556043403578133744722, −6.47806916425287717756983414324, −6.24704939071819496207087011450, −5.72258534851055428004219357605, −5.68636237895206756386659361644, −5.38156402585809942558652171557, −5.37424803847900132745699196753, −5.08360387653250249905864693145, −4.88397518256203805059061476386, −4.28222483882954674924873337418, −3.88873377960504604175660913491, −3.82602350489586755864695107132, −3.58701670280910971974432199198, −2.96816187328029168737610720753, −2.69230425303319982947513031520, −2.43264635136228650574994170717, −2.28707844821845826024212305852, −1.97976735009963365560703596403, −1.60239157221177772127524824236, −1.43612196194275166626716796011, 1.43612196194275166626716796011, 1.60239157221177772127524824236, 1.97976735009963365560703596403, 2.28707844821845826024212305852, 2.43264635136228650574994170717, 2.69230425303319982947513031520, 2.96816187328029168737610720753, 3.58701670280910971974432199198, 3.82602350489586755864695107132, 3.88873377960504604175660913491, 4.28222483882954674924873337418, 4.88397518256203805059061476386, 5.08360387653250249905864693145, 5.37424803847900132745699196753, 5.38156402585809942558652171557, 5.68636237895206756386659361644, 5.72258534851055428004219357605, 6.24704939071819496207087011450, 6.47806916425287717756983414324, 6.49010363556043403578133744722, 6.84586009112041544798600984082, 6.87986371331111101315767090232, 7.23632067042690379541667706123, 7.37866297649901416844393703707, 7.67823653668736374535779847882

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

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