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Properties

Label 12-6042e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.865\times 10^{22}$
Sign $1$
Analytic cond. $1.26109\times 10^{10}$
Root an. cond. $6.94590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 6·3-s + 21·4-s − 8·5-s + 36·6-s − 6·7-s + 56·8-s + 21·9-s − 48·10-s + 3·11-s + 126·12-s − 13·13-s − 36·14-s − 48·15-s + 126·16-s − 5·17-s + 126·18-s − 6·19-s − 168·20-s − 36·21-s + 18·22-s − 12·23-s + 336·24-s + 15·25-s − 78·26-s + 56·27-s − 126·28-s + ⋯
L(s)  = 1  + 4.24·2-s + 3.46·3-s + 21/2·4-s − 3.57·5-s + 14.6·6-s − 2.26·7-s + 19.7·8-s + 7·9-s − 15.1·10-s + 0.904·11-s + 36.3·12-s − 3.60·13-s − 9.62·14-s − 12.3·15-s + 63/2·16-s − 1.21·17-s + 29.6·18-s − 1.37·19-s − 37.5·20-s − 7.85·21-s + 3.83·22-s − 2.50·23-s + 68.5·24-s + 3·25-s − 15.2·26-s + 10.7·27-s − 23.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6} \cdot 53^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6} \cdot 53^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 19^{6} \cdot 53^{6}\)
Sign: $1$
Analytic conductor: \(1.26109\times 10^{10}\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 19^{6} \cdot 53^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
3 \( ( 1 - T )^{6} \)
19 \( ( 1 + T )^{6} \)
53 \( ( 1 + T )^{6} \)
good5 \( 1 + 8 T + 49 T^{2} + 209 T^{3} + 742 T^{4} + 2129 T^{5} + 5222 T^{6} + 2129 p T^{7} + 742 p^{2} T^{8} + 209 p^{3} T^{9} + 49 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 37 T^{2} + 23 p T^{3} + 610 T^{4} + 2007 T^{5} + 5568 T^{6} + 2007 p T^{7} + 610 p^{2} T^{8} + 23 p^{4} T^{9} + 37 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 42 T^{2} - 168 T^{3} + 859 T^{4} - 3607 T^{5} + 11436 T^{6} - 3607 p T^{7} + 859 p^{2} T^{8} - 168 p^{3} T^{9} + 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + p T + 114 T^{2} + 746 T^{3} + 309 p T^{4} + 18145 T^{5} + 70888 T^{6} + 18145 p T^{7} + 309 p^{3} T^{8} + 746 p^{3} T^{9} + 114 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
17 \( 1 + 5 T + 60 T^{2} + 274 T^{3} + 1675 T^{4} + 7673 T^{5} + 32736 T^{6} + 7673 p T^{7} + 1675 p^{2} T^{8} + 274 p^{3} T^{9} + 60 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 121 T^{2} + 929 T^{3} + 6032 T^{4} + 34593 T^{5} + 175436 T^{6} + 34593 p T^{7} + 6032 p^{2} T^{8} + 929 p^{3} T^{9} + 121 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T + 117 T^{2} + 519 T^{3} + 5570 T^{4} + 17518 T^{5} + 175252 T^{6} + 17518 p T^{7} + 5570 p^{2} T^{8} + 519 p^{3} T^{9} + 117 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 17 T + 278 T^{2} + 2823 T^{3} + 25911 T^{4} + 180758 T^{5} + 1131684 T^{6} + 180758 p T^{7} + 25911 p^{2} T^{8} + 2823 p^{3} T^{9} + 278 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 15 T + 206 T^{2} + 1494 T^{3} + 10791 T^{4} + 48459 T^{5} + 328340 T^{6} + 48459 p T^{7} + 10791 p^{2} T^{8} + 1494 p^{3} T^{9} + 206 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 253 T^{2} + 2095 T^{3} + 25373 T^{4} + 159093 T^{5} + 1375802 T^{6} + 159093 p T^{7} + 25373 p^{2} T^{8} + 2095 p^{3} T^{9} + 253 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 4 T + 213 T^{2} + 633 T^{3} + 20088 T^{4} + 46213 T^{5} + 1100572 T^{6} + 46213 p T^{7} + 20088 p^{2} T^{8} + 633 p^{3} T^{9} + 213 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 29 T + 547 T^{2} + 7432 T^{3} + 81225 T^{4} + 723457 T^{5} + 5427798 T^{6} + 723457 p T^{7} + 81225 p^{2} T^{8} + 7432 p^{3} T^{9} + 547 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T + 325 T^{2} - 2181 T^{3} + 44044 T^{4} - 228320 T^{5} + 3345116 T^{6} - 228320 p T^{7} + 44044 p^{2} T^{8} - 2181 p^{3} T^{9} + 325 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 16 T + 251 T^{2} + 1349 T^{3} + 5849 T^{4} - 90945 T^{5} - 692634 T^{6} - 90945 p T^{7} + 5849 p^{2} T^{8} + 1349 p^{3} T^{9} + 251 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 8 T + 55 T^{2} + 57 T^{3} + 6436 T^{4} + 40449 T^{5} + 546624 T^{6} + 40449 p T^{7} + 6436 p^{2} T^{8} + 57 p^{3} T^{9} + 55 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 2 T + 218 T^{2} - 210 T^{3} + 24127 T^{4} - 52900 T^{5} + 1968492 T^{6} - 52900 p T^{7} + 24127 p^{2} T^{8} - 210 p^{3} T^{9} + 218 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 17 T + 246 T^{2} + 2732 T^{3} + 27179 T^{4} + 243467 T^{5} + 2203628 T^{6} + 243467 p T^{7} + 27179 p^{2} T^{8} + 2732 p^{3} T^{9} + 246 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 31 T + 740 T^{2} + 12130 T^{3} + 166991 T^{4} + 1863487 T^{5} + 18081240 T^{6} + 1863487 p T^{7} + 166991 p^{2} T^{8} + 12130 p^{3} T^{9} + 740 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 5 T + 280 T^{2} + 1936 T^{3} + 41907 T^{4} + 285151 T^{5} + 4219320 T^{6} + 285151 p T^{7} + 41907 p^{2} T^{8} + 1936 p^{3} T^{9} + 280 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 5 T + 385 T^{2} + 1459 T^{3} + 71568 T^{4} + 219484 T^{5} + 8022420 T^{6} + 219484 p T^{7} + 71568 p^{2} T^{8} + 1459 p^{3} T^{9} + 385 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 268 T^{2} - 2358 T^{3} + 39909 T^{4} - 386763 T^{5} + 4139276 T^{6} - 386763 p T^{7} + 39909 p^{2} T^{8} - 2358 p^{3} T^{9} + 268 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.46732202456550627045860802214, −4.19552572206949495833578690891, −4.14591264019218556515318035444, −3.98495248264881401635100119295, −3.97479144628820686095711040716, −3.85196016134003305032143923514, −3.69570817650793781393757068405, −3.46219704461771017412730125108, −3.43628738160358189968750448962, −3.38915047522824431299479609001, −3.33886808054631942700402718606, −3.26949681925658433248578387754, −3.25379485596198507570470159119, −2.81452792857746161401735947488, −2.57579457291789323664410357872, −2.53963217597067988118620076603, −2.50777022870816991070765842538, −2.37123258251995189365212460755, −2.32250850032648755311973390431, −1.74630707954889693333306448236, −1.73960403909687261905197458563, −1.66685465757235189752435375195, −1.64894927301031638166461872966, −1.60114781012784567292016207943, −1.38535838386318142775723513102, 0, 0, 0, 0, 0, 0, 1.38535838386318142775723513102, 1.60114781012784567292016207943, 1.64894927301031638166461872966, 1.66685465757235189752435375195, 1.73960403909687261905197458563, 1.74630707954889693333306448236, 2.32250850032648755311973390431, 2.37123258251995189365212460755, 2.50777022870816991070765842538, 2.53963217597067988118620076603, 2.57579457291789323664410357872, 2.81452792857746161401735947488, 3.25379485596198507570470159119, 3.26949681925658433248578387754, 3.33886808054631942700402718606, 3.38915047522824431299479609001, 3.43628738160358189968750448962, 3.46219704461771017412730125108, 3.69570817650793781393757068405, 3.85196016134003305032143923514, 3.97479144628820686095711040716, 3.98495248264881401635100119295, 4.14591264019218556515318035444, 4.19552572206949495833578690891, 4.46732202456550627045860802214

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

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