Properties

Label 12-1680e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.248\times 10^{19}$
Sign $1$
Analytic cond. $5.82798\times 10^{6}$
Root an. cond. $3.66263$
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 − 3·5-s + 2·7-s + 3·9-s − 11-s + 2·13-s − 9·15-s − 8·17-s − 19-s + 6·21-s + 9·23-s + 3·25-s − 2·27-s + 4·31-s − 3·33-s − 6·35-s − 15·37-s + 6·39-s + 10·41-s + 4·43-s − 9·45-s − 11·47-s + 4·49-s − 24·51-s + 53-s + 3·55-s − 3·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s + 0.755·7-s + 9-s − 0.301·11-s + 0.554·13-s − 2.32·15-s − 1.94·17-s − 0.229·19-s + 1.30·21-s + 1.87·23-s + 3/5·25-s − 0.384·27-s + 0.718·31-s − 0.522·33-s − 1.01·35-s − 2.46·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s − 1.34·45-s − 1.60·47-s + 4/7·49-s − 3.36·51-s + 0.137·53-s + 0.404·55-s − 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.82798\times 10^{6}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.989604744\)
\(L(\frac12)\) \(\approx\) \(1.989604744\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 - 2 T + 8 T^{3} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good11 \( 1 + T - 10 T^{2} - 61 T^{3} - 36 T^{4} + 25 p T^{5} + 2506 T^{6} + 25 p^{2} T^{7} - 36 p^{2} T^{8} - 61 p^{3} T^{9} - 10 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - T + 24 T^{2} - 23 T^{3} + 24 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 8 T + 7 T^{2} - 40 T^{3} + 296 T^{4} + 944 T^{5} - 2399 T^{6} + 944 p T^{7} + 296 p^{2} T^{8} - 40 p^{3} T^{9} + 7 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + T + 5 T^{2} - 6 T^{3} + 91 T^{4} + 509 T^{5} + 13054 T^{6} + 509 p T^{7} + 91 p^{2} T^{8} - 6 p^{3} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 42 T^{2} - 135 T^{3} - 348 T^{4} + 7173 T^{5} - 41834 T^{6} + 7173 p T^{7} - 348 p^{2} T^{8} - 135 p^{3} T^{9} + 42 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 9 T^{2} - 256 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \)
31 \( 1 - 4 T - 58 T^{2} + 132 T^{3} + 2326 T^{4} - 1328 T^{5} - 79754 T^{6} - 1328 p T^{7} + 2326 p^{2} T^{8} + 132 p^{3} T^{9} - 58 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 15 T + 117 T^{2} + 442 T^{3} - 1167 T^{4} - 35733 T^{5} - 291366 T^{6} - 35733 p T^{7} - 1167 p^{2} T^{8} + 442 p^{3} T^{9} + 117 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 5 T + 21 T^{2} + 208 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 2 T + 108 T^{2} - 136 T^{3} + 108 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 11 T - 36 T^{2} - 269 T^{3} + 6292 T^{4} + 18647 T^{5} - 222142 T^{6} + 18647 p T^{7} + 6292 p^{2} T^{8} - 269 p^{3} T^{9} - 36 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - T - 38 T^{2} - 835 T^{3} - 148 T^{4} + 16847 T^{5} + 372736 T^{6} + 16847 p T^{7} - 148 p^{2} T^{8} - 835 p^{3} T^{9} - 38 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T - 3 T^{2} + 60 T^{3} + 1992 T^{4} + 32316 T^{5} - 533693 T^{6} + 32316 p T^{7} + 1992 p^{2} T^{8} + 60 p^{3} T^{9} - 3 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 111 T^{2} - 276 T^{3} + 6814 T^{4} + 2692 T^{5} - 435779 T^{6} + 2692 p T^{7} + 6814 p^{2} T^{8} - 276 p^{3} T^{9} - 111 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 10 T - 24 T^{2} - 1656 T^{3} - 7412 T^{4} + 58282 T^{5} + 1219138 T^{6} + 58282 p T^{7} - 7412 p^{2} T^{8} - 1656 p^{3} T^{9} - 24 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 2 T + 67 T^{2} + 304 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 162 T^{2} + 324 T^{3} + 14418 T^{4} - 26244 T^{5} - 1111534 T^{6} - 26244 p T^{7} + 14418 p^{2} T^{8} + 324 p^{3} T^{9} - 162 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 18 T + 30 T^{2} - 28 T^{3} + 15834 T^{4} - 50010 T^{5} - 937026 T^{6} - 50010 p T^{7} + 15834 p^{2} T^{8} - 28 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 4 T + 123 T^{2} + 256 T^{3} + 123 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 22 T + 71 T^{2} + 686 T^{3} + 41412 T^{4} + 291554 T^{5} + 35533 T^{6} + 291554 p T^{7} + 41412 p^{2} T^{8} + 686 p^{3} T^{9} + 71 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 8 T + 251 T^{2} - 1424 T^{3} + 251 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−4.77696245201878202798595363753, −4.58877387988715818033048738077, −4.57239084933477740783611952947, −4.48706361292379366165103062444, −4.12719128693420373751666821986, −4.03700423633928878375821384486, −3.93825892772275191399496670085, −3.81113829906137293701695373592, −3.59416034837587283790702894997, −3.50691588749174051349789040845, −3.46831951472923218814135749913, −2.93078536881886041190941642574, −2.87580559320463063007781340987, −2.76857486163871003613492269088, −2.59781697443951232340216406709, −2.53866803412317864994012970757, −2.46601426494710392619927448136, −2.00709697825968448141800287064, −1.73227612837251201686535677776, −1.72418931001745979105539161450, −1.38793272217856040307081568683, −1.20644071745528630967027350790, −1.00136674719589625973213668074, −0.42956600863031653648731986488, −0.19287980011786603643037384363, 0.19287980011786603643037384363, 0.42956600863031653648731986488, 1.00136674719589625973213668074, 1.20644071745528630967027350790, 1.38793272217856040307081568683, 1.72418931001745979105539161450, 1.73227612837251201686535677776, 2.00709697825968448141800287064, 2.46601426494710392619927448136, 2.53866803412317864994012970757, 2.59781697443951232340216406709, 2.76857486163871003613492269088, 2.87580559320463063007781340987, 2.93078536881886041190941642574, 3.46831951472923218814135749913, 3.50691588749174051349789040845, 3.59416034837587283790702894997, 3.81113829906137293701695373592, 3.93825892772275191399496670085, 4.03700423633928878375821384486, 4.12719128693420373751666821986, 4.48706361292379366165103062444, 4.57239084933477740783611952947, 4.58877387988715818033048738077, 4.77696245201878202798595363753

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

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