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Properties

Label 12-6018e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.750\times 10^{22}$
Sign $1$
Analytic cond. $1.23133\times 10^{10}$
Root an. cond. $6.93209$
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 − 5·5-s − 36·6-s − 7-s + 56·8-s + 21·9-s − 30·10-s − 126·12-s + 2·13-s − 6·14-s + 30·15-s + 126·16-s − 6·17-s + 126·18-s − 105·20-s + 6·21-s − 10·23-s − 336·24-s − 7·25-s + 12·26-s − 56·27-s − 21·28-s − 3·29-s + 180·30-s − 7·31-s + ⋯
L(s)  = 1  + 4.24·2-s − 3.46·3-s + 21/2·4-s − 2.23·5-s − 14.6·6-s − 0.377·7-s + 19.7·8-s + 7·9-s − 9.48·10-s − 36.3·12-s + 0.554·13-s − 1.60·14-s + 7.74·15-s + 63/2·16-s − 1.45·17-s + 29.6·18-s − 23.4·20-s + 1.30·21-s − 2.08·23-s − 68.5·24-s − 7/5·25-s + 2.35·26-s − 10.7·27-s − 3.96·28-s − 0.557·29-s + 32.8·30-s − 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\)
Sign: $1$
Analytic conductor: \(1.23133\times 10^{10}\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
17 \( ( 1 + T )^{6} \)
59 \( ( 1 - T )^{6} \)
good5 \( 1 + p T + 32 T^{2} + 111 T^{3} + 406 T^{4} + 1046 T^{5} + 2714 T^{6} + 1046 p T^{7} + 406 p^{2} T^{8} + 111 p^{3} T^{9} + 32 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + T + 17 T^{2} + 12 T^{3} + 76 T^{4} + 39 T^{5} + 88 T^{6} + 39 p T^{7} + 76 p^{2} T^{8} + 12 p^{3} T^{9} + 17 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 46 T^{2} - 31 T^{3} + 980 T^{4} - 903 T^{5} + 13154 T^{6} - 903 p T^{7} + 980 p^{2} T^{8} - 31 p^{3} T^{9} + 46 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 2 T + 57 T^{2} - 62 T^{3} + 1398 T^{4} - 734 T^{5} + 21480 T^{6} - 734 p T^{7} + 1398 p^{2} T^{8} - 62 p^{3} T^{9} + 57 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 52 T^{2} + 112 T^{3} + 1484 T^{4} + 4352 T^{5} + 32822 T^{6} + 4352 p T^{7} + 1484 p^{2} T^{8} + 112 p^{3} T^{9} + 52 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 + 10 T + 130 T^{2} + 901 T^{3} + 7170 T^{4} + 37455 T^{5} + 215698 T^{6} + 37455 p T^{7} + 7170 p^{2} T^{8} + 901 p^{3} T^{9} + 130 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 54 T^{2} + 165 T^{3} + 2340 T^{4} + 7558 T^{5} + 83874 T^{6} + 7558 p T^{7} + 2340 p^{2} T^{8} + 165 p^{3} T^{9} + 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 7 T + 144 T^{2} + 741 T^{3} + 8602 T^{4} + 35272 T^{5} + 316622 T^{6} + 35272 p T^{7} + 8602 p^{2} T^{8} + 741 p^{3} T^{9} + 144 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 23 T + 378 T^{2} + 4197 T^{3} + 1058 p T^{4} + 292822 T^{5} + 1947662 T^{6} + 292822 p T^{7} + 1058 p^{3} T^{8} + 4197 p^{3} T^{9} + 378 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 172 T^{2} + 1303 T^{3} + 10172 T^{4} + 60169 T^{5} + 399398 T^{6} + 60169 p T^{7} + 10172 p^{2} T^{8} + 1303 p^{3} T^{9} + 172 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 338 T^{2} + 3773 T^{3} + 40422 T^{4} + 318851 T^{5} + 2385318 T^{6} + 318851 p T^{7} + 40422 p^{2} T^{8} + 3773 p^{3} T^{9} + 338 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 290 T^{2} + 2851 T^{3} + 33084 T^{4} + 246157 T^{5} + 2036978 T^{6} + 246157 p T^{7} + 33084 p^{2} T^{8} + 2851 p^{3} T^{9} + 290 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 21 T + 409 T^{2} - 5134 T^{3} + 58832 T^{4} - 521439 T^{5} + 4245260 T^{6} - 521439 p T^{7} + 58832 p^{2} T^{8} - 5134 p^{3} T^{9} + 409 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 2 T + 112 T^{2} + 33 T^{3} + 2922 T^{4} + 1179 T^{5} - 13222 T^{6} + 1179 p T^{7} + 2922 p^{2} T^{8} + 33 p^{3} T^{9} + 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T + 29 T^{2} + 618 T^{3} + 2166 T^{4} - 11169 T^{5} + 481584 T^{6} - 11169 p T^{7} + 2166 p^{2} T^{8} + 618 p^{3} T^{9} + 29 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 4 T + 176 T^{2} - 689 T^{3} + 18988 T^{4} - 85883 T^{5} + 1662882 T^{6} - 85883 p T^{7} + 18988 p^{2} T^{8} - 689 p^{3} T^{9} + 176 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 38 T + 1012 T^{2} + 18237 T^{3} + 264904 T^{4} + 3014553 T^{5} + 28690302 T^{6} + 3014553 p T^{7} + 264904 p^{2} T^{8} + 18237 p^{3} T^{9} + 1012 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 30 T + 578 T^{2} + 8024 T^{3} + 98298 T^{4} + 1047206 T^{5} + 10064370 T^{6} + 1047206 p T^{7} + 98298 p^{2} T^{8} + 8024 p^{3} T^{9} + 578 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 23 T + 456 T^{2} - 6143 T^{3} + 81942 T^{4} - 864052 T^{5} + 8777658 T^{6} - 864052 p T^{7} + 81942 p^{2} T^{8} - 6143 p^{3} T^{9} + 456 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 7 T + 272 T^{2} + 1921 T^{3} + 48268 T^{4} + 284496 T^{5} + 5037542 T^{6} + 284496 p T^{7} + 48268 p^{2} T^{8} + 1921 p^{3} T^{9} + 272 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 10 T + 357 T^{2} + 2104 T^{3} + 46520 T^{4} + 144154 T^{5} + 4147084 T^{6} + 144154 p T^{7} + 46520 p^{2} T^{8} + 2104 p^{3} T^{9} + 357 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.63699858516921735195057270682, −4.48407264617938338355902870558, −4.11154241007131427871083125584, −4.10128606114892096804649344183, −4.03559573524768980972051122883, −4.02194650504328526086455973332, −3.89379341890378324204284993152, −3.50396442621545069952785202938, −3.49966439335480038717597918224, −3.46938638127842162034237895090, −3.45929356379128724664942331574, −3.39353826793180773639116072574, −3.35803920382061382514443120890, −2.74678190034885344848730340648, −2.62787342293924574795379465333, −2.30398747400337014543854952956, −2.29766561128742883722440417668, −2.22622030437138064189540243038, −2.14109722203793144200454221721, −1.63031001963286599863906523773, −1.57117136663077017713117899162, −1.53440484652456537290710192189, −1.46810336408755723883163044601, −1.16978864729094809949450308827, −1.14627897334771595226436455685, 0, 0, 0, 0, 0, 0, 1.14627897334771595226436455685, 1.16978864729094809949450308827, 1.46810336408755723883163044601, 1.53440484652456537290710192189, 1.57117136663077017713117899162, 1.63031001963286599863906523773, 2.14109722203793144200454221721, 2.22622030437138064189540243038, 2.29766561128742883722440417668, 2.30398747400337014543854952956, 2.62787342293924574795379465333, 2.74678190034885344848730340648, 3.35803920382061382514443120890, 3.39353826793180773639116072574, 3.45929356379128724664942331574, 3.46938638127842162034237895090, 3.49966439335480038717597918224, 3.50396442621545069952785202938, 3.89379341890378324204284993152, 4.02194650504328526086455973332, 4.03559573524768980972051122883, 4.10128606114892096804649344183, 4.11154241007131427871083125584, 4.48407264617938338355902870558, 4.63699858516921735195057270682

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

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