Properties

Label 12-912e6-1.1-c1e6-0-11
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  − 3·3-s − 2·5-s + 3·9-s − 3·13-s + 6·15-s − 2·17-s − 4·19-s + 12·23-s + 7·25-s + 2·27-s + 6·29-s + 2·31-s + 9·39-s − 12·41-s − 33·43-s − 6·45-s − 18·47-s + 17·49-s + 6·51-s + 36·53-s + 12·57-s + 2·59-s + 3·61-s + 6·65-s + 19·67-s − 36·69-s − 16·71-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 9-s − 0.832·13-s + 1.54·15-s − 0.485·17-s − 0.917·19-s + 2.50·23-s + 7/5·25-s + 0.384·27-s + 1.11·29-s + 0.359·31-s + 1.44·39-s − 1.87·41-s − 5.03·43-s − 0.894·45-s − 2.62·47-s + 17/7·49-s + 0.840·51-s + 4.94·53-s + 1.58·57-s + 0.260·59-s + 0.384·61-s + 0.744·65-s + 2.32·67-s − 4.33·69-s − 1.89·71-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.363165914\)
\(L(\frac12)\) \(\approx\) \(1.363165914\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T + T^{2} )^{3} \)
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

−5.48245425682134070747435756692, −5.08306015659375717262049470041, −4.95525206620629896070564049006, −4.92971500856206461736936925461, −4.78636697960137146329752975074, −4.54807669791993560211434758134, −4.50071086320388438595832243783, −4.47641183444139684971238358788, −3.95814579391137450980945153486, −3.88905690458774376704939837867, −3.57607587867932585985733948714, −3.44137324420287571874221107412, −3.34130265092125179575627573299, −3.27092515114420637748436822852, −2.80040300193354369364332505204, −2.76438800254682911411037354393, −2.51939945334639758462380931545, −2.21617769611382779068480665309, −2.12294335117106953042883536396, −1.72725514340853779718132374066, −1.46612673850837671748235052364, −1.24055546689813708741794607375, −0.69779571208406653154871227506, −0.54913175304252410757909833278, −0.42692747764386031043589623778, 0.42692747764386031043589623778, 0.54913175304252410757909833278, 0.69779571208406653154871227506, 1.24055546689813708741794607375, 1.46612673850837671748235052364, 1.72725514340853779718132374066, 2.12294335117106953042883536396, 2.21617769611382779068480665309, 2.51939945334639758462380931545, 2.76438800254682911411037354393, 2.80040300193354369364332505204, 3.27092515114420637748436822852, 3.34130265092125179575627573299, 3.44137324420287571874221107412, 3.57607587867932585985733948714, 3.88905690458774376704939837867, 3.95814579391137450980945153486, 4.47641183444139684971238358788, 4.50071086320388438595832243783, 4.54807669791993560211434758134, 4.78636697960137146329752975074, 4.92971500856206461736936925461, 4.95525206620629896070564049006, 5.08306015659375717262049470041, 5.48245425682134070747435756692

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

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