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Properties

Label 12-2736e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.195\times 10^{20}$
Sign $1$
Analytic cond. $1.08732\times 10^{8}$
Root an. cond. $4.67408$
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  + 2·5-s − 3·13-s + 2·17-s − 4·19-s − 12·23-s + 7·25-s − 6·29-s + 2·31-s + 12·41-s − 33·43-s + 18·47-s + 17·49-s − 36·53-s − 2·59-s + 3·61-s − 6·65-s + 19·67-s + 16·71-s + 11·73-s + 9·79-s + 4·85-s − 6·89-s − 8·95-s − 24·97-s + 12·101-s + 30·103-s − 12·109-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.832·13-s + 0.485·17-s − 0.917·19-s − 2.50·23-s + 7/5·25-s − 1.11·29-s + 0.359·31-s + 1.87·41-s − 5.03·43-s + 2.62·47-s + 17/7·49-s − 4.94·53-s − 0.260·59-s + 0.384·61-s − 0.744·65-s + 2.32·67-s + 1.89·71-s + 1.28·73-s + 1.01·79-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 2.43·97-s + 1.19·101-s + 2.95·103-s − 1.14·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.08732\times 10^{8}\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6854105601\)
\(L(\frac12)\) \(\approx\) \(0.6854105601\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + 4 T + 17 T^{2} + 136 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - 2 T - 3 T^{2} + 2 T^{3} - 2 T^{4} + 34 T^{5} - 31 T^{6} + 34 p T^{7} - 2 p^{2} T^{8} + 2 p^{3} T^{9} - 3 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 17 T^{2} + 198 T^{4} - 1549 T^{6} + 198 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 50 T^{2} + 1175 T^{4} - 16364 T^{6} + 1175 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 3 T + 19 T^{2} + 48 T^{3} + 33 T^{4} - 747 T^{5} - 1498 T^{6} - 747 p T^{7} + 33 p^{2} T^{8} + 48 p^{3} T^{9} + 19 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T - 27 T^{2} + 26 T^{3} + 346 T^{4} + 166 T^{5} - 5731 T^{6} + 166 p T^{7} + 346 p^{2} T^{8} + 26 p^{3} T^{9} - 27 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2666 T^{4} + 10704 T^{5} + 45865 T^{6} + 10704 p T^{7} + 2666 p^{2} T^{8} + 588 p^{3} T^{9} + 97 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 79 T^{2} + 402 T^{3} + 3290 T^{4} + 19878 T^{5} + 117439 T^{6} + 19878 p T^{7} + 3290 p^{2} T^{8} + 402 p^{3} T^{9} + 79 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - T + 84 T^{2} - 65 T^{3} + 84 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 5 T^{2} + 3390 T^{4} - 9793 T^{6} + 3390 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 12 T + 75 T^{2} - 324 T^{3} + 102 T^{4} + 16932 T^{5} - 183841 T^{6} + 16932 p T^{7} + 102 p^{2} T^{8} - 324 p^{3} T^{9} + 75 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 33 T + 585 T^{2} + 7326 T^{3} + 71847 T^{4} + 587001 T^{5} + 4129198 T^{6} + 587001 p T^{7} + 71847 p^{2} T^{8} + 7326 p^{3} T^{9} + 585 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 157 T^{2} - 882 T^{3} + 1466 T^{4} + 28590 T^{5} - 316043 T^{6} + 28590 p T^{7} + 1466 p^{2} T^{8} - 882 p^{3} T^{9} + 157 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 36 T + 727 T^{2} + 10620 T^{3} + 123086 T^{4} + 1169736 T^{5} + 9287371 T^{6} + 1169736 p T^{7} + 123086 p^{2} T^{8} + 10620 p^{3} T^{9} + 727 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T - 165 T^{2} - 110 T^{3} + 18142 T^{4} + 4934 T^{5} - 1238089 T^{6} + 4934 p T^{7} + 18142 p^{2} T^{8} - 110 p^{3} T^{9} - 165 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 33 T^{2} - 992 T^{3} + 501 T^{4} + 22395 T^{5} + 500358 T^{6} + 22395 p T^{7} + 501 p^{2} T^{8} - 992 p^{3} T^{9} - 33 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 19 T + 77 T^{2} - 102 T^{3} + 11335 T^{4} - 113975 T^{5} + 583654 T^{6} - 113975 p T^{7} + 11335 p^{2} T^{8} - 102 p^{3} T^{9} + 77 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 16 T + 3 T^{2} + 400 T^{3} + 10462 T^{4} - 68512 T^{5} - 173137 T^{6} - 68512 p T^{7} + 10462 p^{2} T^{8} + 400 p^{3} T^{9} + 3 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 11 T - 53 T^{2} + 932 T^{3} + 1921 T^{4} - 26777 T^{5} - 51994 T^{6} - 26777 p T^{7} + 1921 p^{2} T^{8} + 932 p^{3} T^{9} - 53 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T - 99 T^{2} + 446 T^{3} + 8883 T^{4} + 13851 T^{5} - 1011954 T^{6} + 13851 p T^{7} + 8883 p^{2} T^{8} + 446 p^{3} T^{9} - 99 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 366 T^{2} + 63975 T^{4} - 6705124 T^{6} + 63975 p^{2} T^{8} - 366 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 6 T + 139 T^{2} + 762 T^{3} + 8570 T^{4} + 190218 T^{5} + 1098619 T^{6} + 190218 p T^{7} + 8570 p^{2} T^{8} + 762 p^{3} T^{9} + 139 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 24 T + 499 T^{2} + 7368 T^{3} + 98262 T^{4} + 1070952 T^{5} + 11457671 T^{6} + 1070952 p T^{7} + 98262 p^{2} T^{8} + 7368 p^{3} T^{9} + 499 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.64300864439135456158848312423, −4.47360193964059414280957942152, −4.44206956973426094927649550065, −3.94268849020671384892523421504, −3.92203455467833605760282758309, −3.78420691725536498431461479310, −3.68187050985239033307522051692, −3.53664552375241879507488428782, −3.45421718624740848874046114061, −3.40097958188640066301034354499, −2.91603405591717796267047030212, −2.80586568202150164814110028928, −2.76083049711923952904135286663, −2.44145995979417562704916976541, −2.25025369873161609108147832694, −2.19810311902091220066284625256, −2.03826162957323264810365632690, −2.00563860365815473217296616193, −1.79406449412410976103469972247, −1.32716687998824924181914844926, −1.24030672668590158831006939303, −1.11335211582191243568453988943, −0.828133369157619048612547560271, −0.35746399625445419638985699504, −0.11304189279384620824298217223, 0.11304189279384620824298217223, 0.35746399625445419638985699504, 0.828133369157619048612547560271, 1.11335211582191243568453988943, 1.24030672668590158831006939303, 1.32716687998824924181914844926, 1.79406449412410976103469972247, 2.00563860365815473217296616193, 2.03826162957323264810365632690, 2.19810311902091220066284625256, 2.25025369873161609108147832694, 2.44145995979417562704916976541, 2.76083049711923952904135286663, 2.80586568202150164814110028928, 2.91603405591717796267047030212, 3.40097958188640066301034354499, 3.45421718624740848874046114061, 3.53664552375241879507488428782, 3.68187050985239033307522051692, 3.78420691725536498431461479310, 3.92203455467833605760282758309, 3.94268849020671384892523421504, 4.44206956973426094927649550065, 4.47360193964059414280957942152, 4.64300864439135456158848312423

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

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