Properties

Label 12-8470e6-1.1-c1e6-0-11
Degree $12$
Conductor $3.692\times 10^{23}$
Sign $1$
Analytic cond. $9.57112\times 10^{10}$
Root an. cond. $8.22394$
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 − 3-s + 21·4-s + 6·5-s − 6·6-s − 6·7-s + 56·8-s − 9·9-s + 36·10-s − 21·12-s − 6·13-s − 36·14-s − 6·15-s + 126·16-s − 21·17-s − 54·18-s + 3·19-s + 126·20-s + 6·21-s − 10·23-s − 56·24-s + 21·25-s − 36·26-s + 9·27-s − 126·28-s − 10·29-s − 36·30-s + ⋯
L(s)  = 1  + 4.24·2-s − 0.577·3-s + 21/2·4-s + 2.68·5-s − 2.44·6-s − 2.26·7-s + 19.7·8-s − 3·9-s + 11.3·10-s − 6.06·12-s − 1.66·13-s − 9.62·14-s − 1.54·15-s + 63/2·16-s − 5.09·17-s − 12.7·18-s + 0.688·19-s + 28.1·20-s + 1.30·21-s − 2.08·23-s − 11.4·24-s + 21/5·25-s − 7.06·26-s + 1.73·27-s − 23.8·28-s − 1.85·29-s − 6.57·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.57112\times 10^{10}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 - T )^{6} \)
5 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( 1 + T + 10 T^{2} + 10 T^{3} + 53 T^{4} + 2 p^{3} T^{5} + 191 T^{6} + 2 p^{4} T^{7} + 53 p^{2} T^{8} + 10 p^{3} T^{9} + 10 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 50 T^{2} + 176 T^{3} + 1011 T^{4} + 3002 T^{5} + 15296 T^{6} + 3002 p T^{7} + 1011 p^{2} T^{8} + 176 p^{3} T^{9} + 50 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 21 T + 240 T^{2} + 1936 T^{3} + 12445 T^{4} + 66454 T^{5} + 298845 T^{6} + 66454 p T^{7} + 12445 p^{2} T^{8} + 1936 p^{3} T^{9} + 240 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 58 T^{2} - 136 T^{3} + 1811 T^{4} - 3228 T^{5} + 39719 T^{6} - 3228 p T^{7} + 1811 p^{2} T^{8} - 136 p^{3} T^{9} + 58 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 10 T + 130 T^{2} + 838 T^{3} + 6495 T^{4} + 31660 T^{5} + 186140 T^{6} + 31660 p T^{7} + 6495 p^{2} T^{8} + 838 p^{3} T^{9} + 130 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 10 T + 72 T^{2} + 376 T^{3} + 2819 T^{4} + 12186 T^{5} + 66500 T^{6} + 12186 p T^{7} + 2819 p^{2} T^{8} + 376 p^{3} T^{9} + 72 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 160 T^{2} + 586 T^{3} + 11463 T^{4} + 34898 T^{5} + 462932 T^{6} + 34898 p T^{7} + 11463 p^{2} T^{8} + 586 p^{3} T^{9} + 160 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 2 T + 114 T^{2} - 176 T^{3} + 5483 T^{4} - 24382 T^{5} + 205712 T^{6} - 24382 p T^{7} + 5483 p^{2} T^{8} - 176 p^{3} T^{9} + 114 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 7 T + 112 T^{2} + 822 T^{3} + 7791 T^{4} + 44086 T^{5} + 384331 T^{6} + 44086 p T^{7} + 7791 p^{2} T^{8} + 822 p^{3} T^{9} + 112 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 19 T + 336 T^{2} + 3874 T^{3} + 39637 T^{4} + 323308 T^{5} + 2328247 T^{6} + 323308 p T^{7} + 39637 p^{2} T^{8} + 3874 p^{3} T^{9} + 336 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 10 T + 258 T^{2} - 1926 T^{3} + 27727 T^{4} - 161404 T^{5} + 1674844 T^{6} - 161404 p T^{7} + 27727 p^{2} T^{8} - 1926 p^{3} T^{9} + 258 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 16 T + 276 T^{2} + 2566 T^{3} + 27227 T^{4} + 189042 T^{5} + 1640012 T^{6} + 189042 p T^{7} + 27227 p^{2} T^{8} + 2566 p^{3} T^{9} + 276 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 3 T + 202 T^{2} + 872 T^{3} + 19615 T^{4} + 97768 T^{5} + 1318123 T^{6} + 97768 p T^{7} + 19615 p^{2} T^{8} + 872 p^{3} T^{9} + 202 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 38 T^{2} - 78 T^{3} + 12631 T^{4} - 36014 T^{5} - 373224 T^{6} - 36014 p T^{7} + 12631 p^{2} T^{8} - 78 p^{3} T^{9} - 38 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 27 T + 574 T^{2} + 7848 T^{3} + 94627 T^{4} + 894694 T^{5} + 8025073 T^{6} + 894694 p T^{7} + 94627 p^{2} T^{8} + 7848 p^{3} T^{9} + 574 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 4 T + 160 T^{2} - 1322 T^{3} + 13455 T^{4} - 181366 T^{5} + 841628 T^{6} - 181366 p T^{7} + 13455 p^{2} T^{8} - 1322 p^{3} T^{9} + 160 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 13 T + 366 T^{2} + 4142 T^{3} + 60233 T^{4} + 566182 T^{5} + 5665293 T^{6} + 566182 p T^{7} + 60233 p^{2} T^{8} + 4142 p^{3} T^{9} + 366 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 188 T^{2} + 1612 T^{3} + 20923 T^{4} + 200594 T^{5} + 2287972 T^{6} + 200594 p T^{7} + 20923 p^{2} T^{8} + 1612 p^{3} T^{9} + 188 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 51 T + 1396 T^{2} + 26848 T^{3} + 401083 T^{4} + 4854644 T^{5} + 48532963 T^{6} + 4854644 p T^{7} + 401083 p^{2} T^{8} + 26848 p^{3} T^{9} + 1396 p^{4} T^{10} + 51 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - T + 162 T^{2} + 464 T^{3} + 9129 T^{4} + 190140 T^{5} + 379175 T^{6} + 190140 p T^{7} + 9129 p^{2} T^{8} + 464 p^{3} T^{9} + 162 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 340 T^{2} - 550 T^{3} + 40073 T^{4} + 165172 T^{5} + 3201789 T^{6} + 165172 p T^{7} + 40073 p^{2} T^{8} - 550 p^{3} T^{9} + 340 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.46545907570758899360160493656, −4.15792849217948105169475698394, −4.12263118232089893232358714677, −3.90017885578260835849605718126, −3.86725054454041746865647200102, −3.76413768958125897383789855088, −3.75705404804887472767793536280, −3.23558749106915858518544225028, −3.08033679931687761788169513508, −3.05232054536707950678398716086, −3.04976962692796330837463404373, −3.02493676580132302758944729485, −3.01892201642187531320975276313, −2.50016783508527253878429473649, −2.47807145665829558945237704441, −2.35137843060326188664069937509, −2.32306475798589800722855126549, −2.31462741553372275049346561968, −2.14818008107017216321183840285, −1.85360265533893013204759998280, −1.84851509265254020088176189606, −1.45306128597451646460832657660, −1.33222352711768471800831547378, −1.30973733228809207115518102709, −1.20750080573714985619795575227, 0, 0, 0, 0, 0, 0, 1.20750080573714985619795575227, 1.30973733228809207115518102709, 1.33222352711768471800831547378, 1.45306128597451646460832657660, 1.84851509265254020088176189606, 1.85360265533893013204759998280, 2.14818008107017216321183840285, 2.31462741553372275049346561968, 2.32306475798589800722855126549, 2.35137843060326188664069937509, 2.47807145665829558945237704441, 2.50016783508527253878429473649, 3.01892201642187531320975276313, 3.02493676580132302758944729485, 3.04976962692796330837463404373, 3.05232054536707950678398716086, 3.08033679931687761788169513508, 3.23558749106915858518544225028, 3.75705404804887472767793536280, 3.76413768958125897383789855088, 3.86725054454041746865647200102, 3.90017885578260835849605718126, 4.12263118232089893232358714677, 4.15792849217948105169475698394, 4.46545907570758899360160493656

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

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