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Properties

Label 12-2646e6-1.1-c1e6-0-6
Degree $12$
Conductor $3.432\times 10^{20}$
Sign $1$
Analytic cond. $8.89614\times 10^{7}$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s − 5-s + 2·8-s + 3·10-s + 11-s + 8·13-s − 9·16-s − 8·17-s + 6·19-s − 3·20-s − 3·22-s + 7·23-s + 9·25-s − 24·26-s + 5·29-s + 20·31-s + 9·32-s + 24·34-s − 6·37-s − 18·38-s − 2·40-s − 6·43-s + 3·44-s − 21·46-s + 9·47-s − 27·50-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.447·5-s + 0.707·8-s + 0.948·10-s + 0.301·11-s + 2.21·13-s − 9/4·16-s − 1.94·17-s + 1.37·19-s − 0.670·20-s − 0.639·22-s + 1.45·23-s + 9/5·25-s − 4.70·26-s + 0.928·29-s + 3.59·31-s + 1.59·32-s + 4.11·34-s − 0.986·37-s − 2.91·38-s − 0.316·40-s − 0.914·43-s + 0.452·44-s − 3.09·46-s + 1.31·47-s − 3.81·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(8.89614\times 10^{7}\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.914780137\)
\(L(\frac12)\) \(\approx\) \(4.914780137\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 52 p T^{7} + 23 p^{2} T^{8} - 17 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T - 26 T^{2} + 23 T^{3} + 37 p T^{4} - 202 T^{5} - 4853 T^{6} - 202 p T^{7} + 37 p^{3} T^{8} + 23 p^{3} T^{9} - 26 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 4 T + 39 T^{2} + 112 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 3946 p T^{7} + 2423 p^{2} T^{8} + 83 p^{3} T^{9} - 32 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + T + p T^{2} )^{6} \)
41 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 15 T + 225 T^{2} - 1671 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 35126 p T^{7} + 11666 p^{2} T^{8} + 154 p^{3} T^{9} - 20 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 19 T + 227 T^{2} + 2143 T^{3} + 227 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 13564 p T^{7} + 18788 p^{2} T^{8} + 2 p^{3} T^{9} - 182 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 9 T + 225 T^{2} + 1611 T^{3} + 225 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.59828607609191154346685054198, −4.53701529906230454888908028261, −4.39411818509083939274297372996, −4.34218231617599454944520556693, −3.90485605118626574547609901547, −3.83226101871161574634998952791, −3.60329275111425728005065141886, −3.50412620146750151712427142982, −3.32206102315913919660356818385, −3.23325572092627455370510861153, −3.14138220163184514049850177015, −2.84848826350845918595296649361, −2.69909173961159426220167696872, −2.43306836181327659181907478132, −2.34493898517393204812768441220, −2.20928941181920677413335479916, −1.97045152944271592003250405072, −1.60549578760723316103787683463, −1.51496702178083180627858624362, −1.38924468726173350089594343242, −0.887267577064356942672359299640, −0.815376185556146727469851579262, −0.76076830674428686830131236024, −0.69472788882364192771843382213, −0.47542491547167553996043021978, 0.47542491547167553996043021978, 0.69472788882364192771843382213, 0.76076830674428686830131236024, 0.815376185556146727469851579262, 0.887267577064356942672359299640, 1.38924468726173350089594343242, 1.51496702178083180627858624362, 1.60549578760723316103787683463, 1.97045152944271592003250405072, 2.20928941181920677413335479916, 2.34493898517393204812768441220, 2.43306836181327659181907478132, 2.69909173961159426220167696872, 2.84848826350845918595296649361, 3.14138220163184514049850177015, 3.23325572092627455370510861153, 3.32206102315913919660356818385, 3.50412620146750151712427142982, 3.60329275111425728005065141886, 3.83226101871161574634998952791, 3.90485605118626574547609901547, 4.34218231617599454944520556693, 4.39411818509083939274297372996, 4.53701529906230454888908028261, 4.59828607609191154346685054198

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

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